#set document(title: "10.5 Graphing Quadratic Equations in Two Variables", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 10.5#h(0.6em)Graphing Quadratic Equations in Two Variables #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. Graph the equation #math.equation(block: false, alt: "y equals 3 x minus 5")[$y = 3 x − 5$] by plotting points. #linebreak() If you missed this problem, review . #solutionbox[ #figure(figph[A graph shows a straight line with a positive slope in a Cartesian coordinate system. The line passes through approximately (0, -5), (1, -2), (2, 1), and (3, 4).], alt: "A graph shows a straight line with a positive slope in a Cartesian coordinate system. The line passes through approximately (0, -5), (1, -2), (2, 1), and (3, 4).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate #math.equation(block: false, alt: "2 x squared plus 4 x minus 1")[$2 x^(2) + 4 x − 1$] when #math.equation(block: false, alt: "x equals −3")[$x = −3$]. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "5")[$5$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate #math.equation(block: false, alt: "minus the fraction b over 2 a")[$− frac(b, 2 a)$] when #math.equation(block: false, alt: "a equals the fraction 1 over 3")[$a = frac(1, 3)$] and #math.equation(block: false, alt: "b equals the fraction 5 over 6")[$b = frac(5, 6)$]. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "minus the fraction 5 over 4")[$− frac(5, 4)$] ] ] === Recognize the Graph of a Quadratic Equation in Two Variables We have graphed equations of the form #math.equation(block: false, alt: "A x plus B y equals C")[$A x + B y = C$]. We called equations like this linear equations because their graphs are straight lines. Now, we will graph equations of the form #math.equation(block: false, alt: "y equals a x squared plus b x plus c")[$y = a x^(2) + b x + c$]. We call this kind of equation a #strong[quadratic equation in two variables]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Quadratic Equation in Two Variables] A #strong[quadratic equation in two variables], where #math.equation(block: false, alt: "a , b , and c")[$a , b , #h(0.2em) "and" #h(0.2em) c$] are real numbers and #math.equation(block: false, alt: "a not equal to 0")[$a ≠ 0$], is an equation of the form #math.equation(block: true, alt: "y equals a x squared plus b x plus c")[$y = a x^(2) + b x + c$] ] Just like we started graphing linear equations by plotting points, we will do the same for quadratic equations. Let’s look first at graphing the quadratic equation #math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$]. We will choose integer values of #math.equation(block: false, alt: "x")[$x$] between #math.equation(block: false, alt: "−2")[$−2$] and 2 and find their #math.equation(block: false, alt: "y")[$y$] values. #figure(table( columns: 2, align: left, inset: 6pt, [#strong[#math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$]]], [], [#strong[#math.equation(block: false, alt: "x")[$x$]]], [#strong[#math.equation(block: false, alt: "y")[$y$]]], [0], [0], [1], [1], [#math.equation(block: false, alt: "−1")[$−1$]], [1], [2], [4], [#math.equation(block: false, alt: "−2")[$−2$]], [4], )) Notice when we let #math.equation(block: false, alt: "x equals 1")[$x = 1$] and #math.equation(block: false, alt: "x equals −1")[$x = −1$], we got the same value for #math.equation(block: false, alt: "y")[$y$]. #math.equation(block: true, alt: "y equals x squared, y equals x squared; y equals 1 squared, y equals open parenthesis −1 close parenthesis squared; y equals 1, y equals 1")[$y = x^(2) & & & y = x^(2) \ y = 1^(2) & & & y = attach(( −1 ), t: 2) \ y = 1 & & & y = 1$]The same thing happened when we let #math.equation(block: false, alt: "x equals 2")[$x = 2$] and #math.equation(block: false, alt: "x equals −2")[$x = −2$]. Now, we will plot the points to show the graph of #math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$]. #figure(figph[This figure shows an upward-opening u shaped curve graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The lowest point on the curve is at the point (0, 0). Other points on the curve are located at (-2, 4), (-1, 1), (1, 1) and (2, 4).], alt: "This figure shows an upward-opening u shaped curve graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The lowest point on the curve is at the point (0, 0). Other points on the curve are located at (-2, 4), (-1, 1), (1, 1) and (2, 4).", caption: none) The graph is not a line. This figure is called a #strong[parabola]. Every quadratic equation has a graph that looks like this. In you will practice graphing a parabola by plotting a few points. #examplebox("Example 1")[][ Graph #math.equation(block: false, alt: "y equals x squared minus 1")[$y = x^(2) − 1$]. #solutionbox[ We will graph the equation by plotting points. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Choose integers values for #emph[x], substitute them into the equation and solve for #emph[y].], []), [Record the values of the ordered pairs in the chart.], [#figure(figph[This table illustrates points for the parabola y = x^2 - 1, including its vertex (0, -1) and x-intercepts (1, 0) and (-1, 0).], alt: "This table illustrates points for the parabola y = x^2 - 1, including its vertex (0, -1) and x-intercepts (1, 0) and (-1, 0).", caption: none)], [Plot the points, and then connect them with a smooth curve. The result will be the graph of the equation#math.equation(block: false, alt: "y equals x squared minus 1")[$y = x^(2) − 1$].], [#figure(figph[An upward-opening parabola is plotted on a coordinate plane, with its vertex at (0,-1) and symmetric about the y-axis, indicating a quadratic function.], alt: "An upward-opening parabola is plotted on a coordinate plane, with its vertex at (0,-1) and symmetric about the y-axis, indicating a quadratic function.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph #math.equation(block: false, alt: "y equals − x squared")[$y = "−" x^(2)$]. #solutionbox[ #figure(figph[This figure shows a downward-opening u shaped curve graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The highest point on the curve is at the point (0, 0). Other points on the curve are located at (-2, -4), (-1, -1), (1, -1) and (2, -4).], alt: "This figure shows a downward-opening u shaped curve graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The highest point on the curve is at the point (0, 0). Other points on the curve are located at (-2, -4), (-1, -1), (1, -1) and (2, -4).", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph #math.equation(block: false, alt: "y equals x squared plus 1")[$y = x^(2) + 1$]. #solutionbox[ #figure(figph[This figure shows an upward-opening u shaped curve graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The lowest point on the curve is at the point (0, 1). Other points on the curve are located at (-2, 5), (-1, 2), (1, 2) and (2, 5).], alt: "This figure shows an upward-opening u shaped curve graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The lowest point on the curve is at the point (0, 1). Other points on the curve are located at (-2, 5), (-1, 2), (1, 2) and (2, 5).", caption: none) ] ] How do the equations #math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$] and #math.equation(block: false, alt: "y equals x squared minus 1")[$y = x^(2) − 1$] differ? What is the difference between their graphs? How are their graphs the same? All parabolas of the form #math.equation(block: false, alt: "y equals a x squared plus b x plus c")[$y = a x^(2) + b x + c$] open upwards or downwards. #figure(figph[This figure shows two graphs side by side. The graph on the left side shows an upward-opening u shaped curve graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The lowest point on the curve is at the point (-2, -1). Other points on the curve are located at (-3, 0), and (-1, 0). Below the graph is the equation y equals a squared plus b x plus c. Below that is the equation of the graph, y equals x squared plus 4 x plus 3. Below that is the inequality a greater than 0 which means the parabola opens upwards. The graph on the right side shows a downward-opening u shaped curve graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The highest point on the curve is at the point (2, 7). Other points on the curve are located at (0, 3), and (4, 3). Below the graph is the equation y equals a squared plus b x plus c. Below that is the equation of the graph, y equals negative x squared plus 4 x plus 3. Below that is the inequality a less than 0 which means the parabola opens downwards.], alt: "This figure shows two graphs side by side. The graph on the left side shows an upward-opening u shaped curve graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The lowest point on the curve is at the point (-2, -1). Other points on the curve are located at (-3, 0), and (-1, 0). Below the graph is the equation y equals a squared plus b x plus c. Below that is the equation of the graph, y equals x squared plus 4 x plus 3. Below that is the inequality a greater than 0 which means the parabola opens upwards. The graph on the right side shows a downward-opening u shaped curve graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The highest point on the curve is at the point (2, 7). Other points on the curve are located at (0, 3), and (4, 3). Below the graph is the equation y equals a squared plus b x plus c. Below that is the equation of the graph, y equals negative x squared plus 4 x plus 3. Below that is the inequality a less than 0 which means the parabola opens downwards.", caption: none) Notice that the only difference in the two equations is the negative sign before the #math.equation(block: false, alt: "x squared")[$x^(2)$] in the equation of the second graph in . When the #math.equation(block: false, alt: "x squared")[$x^(2)$] term is positive, the #strong[parabola] opens upward, and when the #math.equation(block: false, alt: "x squared")[$x^(2)$] term is negative, the parabola opens downward. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Parabola Orientation] For the quadratic equation #math.equation(block: false, alt: "y equals a x squared plus b x plus c")[$y = a x^(2) + b x + c$], if: #figure(figph[The image shows two statements. The first statement reads “a greater than 0, the parabola opens upwards”. This statement is followed by the image of an upward opening parabola. The second statement reads “a less than 0, the parabola opens downward”. This statement is followed by the image of a downward opening parabola.], alt: "The image shows two statements. The first statement reads “a greater than 0, the parabola opens upwards”. This statement is followed by the image of an upward opening parabola. The second statement reads “a less than 0, the parabola opens downward”. This statement is followed by the image of a downward opening parabola.", caption: none) ] #examplebox("Example 2")[][ Determine whether each parabola opens upward or downward: ⓐ #math.equation(block: false, alt: "y equals −3 x squared plus 2 x minus 4")[$y = −3 x^(2) + 2 x − 4$] ⓑ #math.equation(block: false, alt: "y equals 6 x squared plus 7 x minus 9")[$y = 6 x^(2) + 7 x − 9$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ #linebreak() Find the value of "#emph[a]".], [#figure(figph[This image displays a comparison between the standard quadratic equation, y = ax^2 + bx + c, and a specific instance, y = -3x^2 + 2x - 4. It highlights that in this case, a = -3.], alt: "This image displays a comparison between the standard quadratic equation, y = ax^2 + bx + c, and a specific instance, y = -3x^2 + 2x - 4. It highlights that in this case, a = -3.", caption: none) #linebreak() Since the “a” is negative, the parabola will open downward.]), [ⓑ #linebreak() Find the value of "#emph[a]".], [#figure(figph[The image displays the general quadratic equation y = ax^2 + bx + c followed by a specific example y = 6x^2 + 7x - 9. Below this, a variable q is assigned the value 6.], alt: "The image displays the general quadratic equation y = ax^2 + bx + c followed by a specific example y = 6x^2 + 7x - 9. Below this, a variable q is assigned the value 6.", caption: none) #linebreak() Since the “a” is positive, the parabola will open upward.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine whether each parabola opens upward or downward: ⓐ #math.equation(block: false, alt: "y equals 2 x squared plus 5 x minus 2")[$y = 2 x^(2) + 5 x − 2$] ⓑ #math.equation(block: false, alt: "y equals −3 x squared minus 4 x plus 7")[$y = −3 x^(2) − 4 x + 7$] #solutionbox[ ⓐ up ⓑ down ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine whether each parabola opens upward or downward: ⓐ #math.equation(block: false, alt: "y equals −2 x squared minus 2 x minus 3")[$y = −2 x^(2) − 2 x − 3$] ⓑ #math.equation(block: false, alt: "y equals 5 x squared minus 2 x minus 1")[$y = 5 x^(2) − 2 x − 1$] #solutionbox[ ⓐ down ⓑ up ] ] === Find the Axis of Symmetry and Vertex of a Parabola Look again at . Do you see that we could fold each parabola in half and that one side would lie on top of the other? The ‘fold line’ is a line of symmetry. We call it the #strong[axis of symmetry] of the parabola. We show the same two graphs again with the axis of symmetry in blue. #figure(figph[This figure shows an two graphs side by side. The graph on the left side shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The lowest point on the curve is at the point (-2, -1). Other points on the curve are located at (-3, 0), and (-1, 0). Also on the graph is a dashed vertical line that goes through the center of the parabola at the point (-2, -1). Below the graph is the equation of the graph, y equals x squared plus 4 x plus 3. The graph on the right side shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The highest point on the curve is at the point (2, 7). Other points on the curve are located at (0, 3), and (4, 3). Also on the graph is a dashed vertical line that goes through the center of the parabola at the point (2, 7). Below the graph is the equation of the graph, y equals negative x squared plus 4 x plus 3.], alt: "This figure shows an two graphs side by side. The graph on the left side shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The lowest point on the curve is at the point (-2, -1). Other points on the curve are located at (-3, 0), and (-1, 0). Also on the graph is a dashed vertical line that goes through the center of the parabola at the point (-2, -1). Below the graph is the equation of the graph, y equals x squared plus 4 x plus 3. The graph on the right side shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The highest point on the curve is at the point (2, 7). Other points on the curve are located at (0, 3), and (4, 3). Also on the graph is a dashed vertical line that goes through the center of the parabola at the point (2, 7). Below the graph is the equation of the graph, y equals negative x squared plus 4 x plus 3.", caption: none) The equation of the #strong[axis of symmetry] can be derived by using the Quadratic Formula. We will omit the derivation here and proceed directly to using the result. The equation of the axis of symmetry of the graph of #math.equation(block: false, alt: "y equals a x squared plus b x plus c")[$y = a x^(2) + b x + c$] is #math.equation(block: false, alt: "x equals minus the fraction b over 2 a .")[$x = − frac(b, 2 a) .$] So, to find the equation of symmetry of each of the parabolas we graphed above, we will substitute into the formula #math.equation(block: false, alt: "x equals minus the fraction b over 2 a")[$x = − frac(b, 2 a)$]. #figure(figph[The figure shows the steps to find the axis of symmetry for two parabolas. On the left side the standard form of a quadratic equation which is y equals a x squared plus b x plus c is written above the given equation y equals x squared plus 4 x plus 3. The axis of symmetry is the equation x equals negative b divided by the quantity two times a. Plugging in the values of a and b from the quadratic equation the formula becomes x equals negative 4 divided by the quantity 2 times 1, which simplifies to x equals negative 2. On the right side the standard form of a quadratic equation which is y equals a x squared plus b x plus c is written above the given equation y equals negative x squared plus 4 x plus 3. The axis of symmetry is the equation x equals negative b divided by the quantity two times a. Plugging in the values of a and b from the quadratic equation the formula becomes x equals negative 4 divided by the quantity 2 times -1, which simplifies to x equals 2.], alt: "The figure shows the steps to find the axis of symmetry for two parabolas. On the left side the standard form of a quadratic equation which is y equals a x squared plus b x plus c is written above the given equation y equals x squared plus 4 x plus 3. The axis of symmetry is the equation x equals negative b divided by the quantity two times a. Plugging in the values of a and b from the quadratic equation the formula becomes x equals negative 4 divided by the quantity 2 times 1, which simplifies to x equals negative 2. On the right side the standard form of a quadratic equation which is y equals a x squared plus b x plus c is written above the given equation y equals negative x squared plus 4 x plus 3. The axis of symmetry is the equation x equals negative b divided by the quantity two times a. Plugging in the values of a and b from the quadratic equation the formula becomes x equals negative 4 divided by the quantity 2 times -1, which simplifies to x equals 2.", caption: none) Look back at . Are these the equations of the dashed red lines? The point on the parabola that is on the axis of symmetry is the lowest or highest point on the parabola, depending on whether the parabola opens upwards or downwards. This point is called the #strong[vertex] of the parabola. We can easily find the coordinates of the vertex, because we know it is on the axis of symmetry. This means its #emph[x]-coordinate is #math.equation(block: false, alt: "minus the fraction b over 2 a")[$− frac(b, 2 a)$]. To find the #emph[y]-coordinate of the vertex, we substitute the value of the #emph[x]-coordinate into the quadratic equation. #figure(figph[The figure shows the steps to find the vertex for two parabolas. On the left side is the given equation y equals x squared plus 4 x plus 3. Below the equation is the statement “axis of symmetry is x equals -2”. Below that is the statement “vertex is” next to the statement is an ordered pair with x-value of -2, the same as the axis of symmetry, and the y-value is blank. Below that the original equation is rewritten. Below the equation is the equation with -2 plugged in for the x value which is y equals -2 squared plus 4 times -2 plus 3. This simplifies to y equals -1. Below this is the statement “vertex is (-2, -1)”. On the right side is the given equation y equals negative x squared plus 4 x plus 3. Below the equation is the statement “axis of symmetry is x equals 2”. Below that is the statement “vertex is” next to the statement is an ordered pair with x-value of 2, the same as the axis of symmetry, and the y-value is blank. Below that the original equation is rewritten. Below the equation is the equation with 2 plugged in for the x value which is y equals negative the quantity 2 squared, plus 4 times 2 plus 3. This simplifies to y equals 7. Below this is the statement “vertex is (2, 7)”.], alt: "The figure shows the steps to find the vertex for two parabolas. On the left side is the given equation y equals x squared plus 4 x plus 3. Below the equation is the statement “axis of symmetry is x equals -2”. Below that is the statement “vertex is” next to the statement is an ordered pair with x-value of -2, the same as the axis of symmetry, and the y-value is blank. Below that the original equation is rewritten. Below the equation is the equation with -2 plugged in for the x value which is y equals -2 squared plus 4 times -2 plus 3. This simplifies to y equals -1. Below this is the statement “vertex is (-2, -1)”. On the right side is the given equation y equals negative x squared plus 4 x plus 3. Below the equation is the statement “axis of symmetry is x equals 2”. Below that is the statement “vertex is” next to the statement is an ordered pair with x-value of 2, the same as the axis of symmetry, and the y-value is blank. Below that the original equation is rewritten. Below the equation is the equation with 2 plugged in for the x value which is y equals negative the quantity 2 squared, plus 4 times 2 plus 3. This simplifies to y equals 7. Below this is the statement “vertex is (2, 7)”.", caption: none) #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Axis of Symmetry and Vertex of a Parabola] For a parabola with equation #math.equation(block: false, alt: "y equals a x squared plus b x plus c")[$y = a x^(2) + b x + c$]: - The axis of symmetry of a parabola is the line #math.equation(block: false, alt: "x equals minus the fraction b over 2 a")[$x = − frac(b, 2 a)$]. - The vertex is on the axis of symmetry, so its #emph[x]-coordinate is #math.equation(block: false, alt: "minus the fraction b over 2 a")[$− frac(b, 2 a)$]. To find the #emph[y]-coordinate of the vertex, we substitute #math.equation(block: false, alt: "x equals minus the fraction b over 2 a")[$x = − frac(b, 2 a)$] into the quadratic equation. ] #examplebox("Example 3")[][ For the parabola #math.equation(block: false, alt: "y equals 3 x squared minus 6 x plus 2")[$y = 3 x^(2) − 6 x + 2$] find: ⓐ the axis of symmetry and ⓑ the vertex. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ], [#figure(figph[The image displays the standard form of a quadratic equation, y = ax^2 + bx + c, followed by a concrete instance, y = 3x^2 - 6x + 2, demonstrating how specific values for a, b, and c are substituted.], alt: "The image displays the standard form of a quadratic equation, y = ax^2 + bx + c, followed by a concrete instance, y = 3x^2 - 6x + 2, demonstrating how specific values for a, b, and c are substituted.", caption: none)]), [The axis of symmetry is the line #math.equation(block: false, alt: "x equals minus the fraction b over 2 a")[$x = − frac(b, 2 a)$].], [#figure(figph[The formula for the x-coordinate of the vertex of a parabola, or the axis of symmetry, derived from the standard quadratic equation: x = -b / (2a).], alt: "The formula for the x-coordinate of the vertex of a parabola, or the axis of symmetry, derived from the standard quadratic equation: x = -b / (2a).", caption: none)], [Substitute the values of #emph[a, b] into the equation.], [#figure(figph[A mathematical equation shows X equals the negative of a fraction, where the numerator is -6 and the denominator is 2 multiplied by 3.], alt: "A mathematical equation shows X equals the negative of a fraction, where the numerator is -6 and the denominator is 2 multiplied by 3.", caption: none)], [Simplify.], [#math.equation(block: false, alt: "x equals 1")[$x = 1$]], [], [The axis of symmetry is the line #math.equation(block: false, alt: "x equals 1")[$x = 1$].], [ⓑ], [#figure(figph[A quadratic equation, y = 3x^2 - 6x + 2, is displayed in black text on a white background.], alt: "A quadratic equation, y = 3x^2 - 6x + 2, is displayed in black text on a white background.", caption: none)], [The vertex is on the line of symmetry, so its #emph[x]-coordinate will be #math.equation(block: false, alt: "x equals 1")[$x = 1$].], [], [Substitute#math.equation(block: false, alt: "x equals 1")[$x = 1$] into the equation and solve for #emph[y.]], [#figure(figph[A mathematical equation shows 'y = 3(1)^2 - 6(1) + 2', demonstrating the substitution of the number 1 into a quadratic expression, with the number 1 highlighted in red for emphasis.], alt: "A mathematical equation shows 'y = 3(1)^2 - 6(1) + 2', demonstrating the substitution of the number 1 into a quadratic expression, with the number 1 highlighted in red for emphasis.", caption: none)], [Simplify.], [#figure(figph[A mathematical equation is displayed, reading y = 3 \* 1 - 6 + 2. The equation is rendered in black text on a white background, representing a calculation involving multiplication, subtraction, and addition.], alt: "A mathematical equation is displayed, reading y = 3 * 1 - 6 + 2. The equation is rendered in black text on a white background, representing a calculation involving multiplication, subtraction, and addition.", caption: none)], [This is the #emph[y]-coordinate.], [#math.equation(block: false, alt: "y equals −1")[$y = −1$] #linebreak() The vertex is #math.equation(block: false, alt: "open parenthesis 1 , − 1 close parenthesis .")[$( 1 , "−" 1 ) .$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ For the parabola #math.equation(block: false, alt: "y equals 2 x squared minus 8 x plus 1")[$y = 2 x^(2) − 8 x + 1$] find: ⓐ the axis of symmetry and ⓑ the vertex. #solutionbox[ ⓐ #math.equation(block: false, alt: "x equals 2")[$x = 2$] ⓑ #math.equation(block: false, alt: "open parenthesis 2 , −7 close parenthesis")[$( 2 , −7 )$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ For the parabola #math.equation(block: false, alt: "y equals 2 x squared minus 4 x minus 3")[$y = 2 x^(2) − 4 x − 3$] find: ⓐ the axis of symmetry and ⓑ the vertex. #solutionbox[ ⓐ #math.equation(block: false, alt: "x equals 1")[$x = 1$] ⓑ #math.equation(block: false, alt: "open parenthesis 1 , −5 close parenthesis")[$( 1 , −5 )$] ] ] === Find the Intercepts of a Parabola When we graphed linear equations, we often used the #emph[x]- and #emph[y]-intercepts to help us graph the lines. Finding the coordinates of the intercepts will help us to graph parabolas, too. Remember, at the #strong[#emph[y]-intercept] the value of #math.equation(block: false, alt: "x")[$x$] is zero. So, to find the #emph[y]-intercept, we substitute #math.equation(block: false, alt: "x equals 0")[$x = 0$] into the equation. Let’s find the #emph[y]-intercepts of the two parabolas shown in the figure below. #figure(figph[This figure shows an two graphs side by side. The graph on the left side shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The vertex is at the point (-2, -1). Other points on the curve are located at (-3, 0), and (-1, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -2. Below the graph is the equation of the graph, y equals x squared plus 4 x plus 3. Below that is the statement “x equals 0”. Next to that is the equation of the graph with 0 plugged in for x which gives y equals 0 squared plus4 times 0 plus 3. This simplifies to y equals 3. Below the equation is the statement “y-intercept (0, 3)”. The graph on the right side shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The vertex is at the point (2, 7). Other points on the curve are located at (0, 3), and (4, 3). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 2. Below the graph is the equation of the graph, y equals negative x squared plus 4 x plus 3. Below that is the statement “x equals 0”. Next to that is the equation of the graph with 0 plugged in for x which gives y equals negative quantity 0 squared plus 4 times 0 plus 3. This simplifies to y equals 3. Below the equation is the statement “y-intercept (0, 3)”.], alt: "This figure shows an two graphs side by side. The graph on the left side shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The vertex is at the point (-2, -1). Other points on the curve are located at (-3, 0), and (-1, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -2. Below the graph is the equation of the graph, y equals x squared plus 4 x plus 3. Below that is the statement “x equals 0”. Next to that is the equation of the graph with 0 plugged in for x which gives y equals 0 squared plus4 times 0 plus 3. This simplifies to y equals 3. Below the equation is the statement “y-intercept (0, 3)”. The graph on the right side shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The vertex is at the point (2, 7). Other points on the curve are located at (0, 3), and (4, 3). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 2. Below the graph is the equation of the graph, y equals negative x squared plus 4 x plus 3. Below that is the statement “x equals 0”. Next to that is the equation of the graph with 0 plugged in for x which gives y equals negative quantity 0 squared plus 4 times 0 plus 3. This simplifies to y equals 3. Below the equation is the statement “y-intercept (0, 3)”.", caption: none) At an #strong[#emph[x]-intercept], the value of #math.equation(block: false, alt: "y")[$y$] is zero. To find an #emph[x]-intercept, we substitute #math.equation(block: false, alt: "y equals 0")[$y = 0$] into the equation. In other words, we will need to solve the equation #math.equation(block: false, alt: "0 equals a x squared plus b x plus c")[$0 = a x^(2) + b x + c$] for #math.equation(block: false, alt: "x")[$x$]. #math.equation(block: true, alt: "y equals a x squared plus b x plus c; 0 equals a x squared plus b x plus c")[$y = a x^(2) + b x + c \ 0 = a x^(2) + b x + c$]But solving quadratic equations like this is exactly what we have done earlier in this chapter. We can now find the #emph[x]-intercepts of the two parabolas shown. First, we will find the #strong[#emph[x]-intercepts of a parabola] with equation #math.equation(block: false, alt: "y equals x squared plus 4 x plus 3")[$y = x^(2) + 4 x + 3$]. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image shows the quadratic equation y = x^2 + 4x + 3.], alt: "The image shows the quadratic equation y = x^2 + 4x + 3.", caption: none)]), [Let #math.equation(block: false, alt: "y equals 0")[$y = 0$].], [#figure(figph[A mathematical equation displays '0 = x^2 + 4x + 3' in black font against a white background, representing a quadratic equation.], alt: "A mathematical equation displays '0 = x^2 + 4x + 3' in black font against a white background, representing a quadratic equation.", caption: none)], [Factor.], [#figure(figph[The equation 0 = (x + 1)(x + 3) is displayed, illustrating a quadratic equation in factored form, ready to be solved for x by setting each factor to zero.], alt: "The equation 0 = (x + 1)(x + 3) is displayed, illustrating a quadratic equation in factored form, ready to be solved for x by setting each factor to zero.", caption: none)], [Use the zero product property.], [#figure(figph[Two mathematical equations are displayed: x+1=0 and x+3=0. These appear to be simple linear equations or factors in an algebraic problem.], alt: "Two mathematical equations are displayed: x+1=0 and x+3=0. These appear to be simple linear equations or factors in an algebraic problem.", caption: none)], [Solve.], [#figure(figph[The image displays the mathematical solutions 'x = -1, x = -3' in a black font against a plain white background, indicating two distinct values for the variable x.], alt: "The image displays the mathematical solutions 'x = -1, x = -3' in a black font against a plain white background, indicating two distinct values for the variable x.", caption: none)], [], [The #emph[x] intercepts are #math.equation(block: false, alt: "open parenthesis − 1 , 0 close parenthesis")[$( "−" 1 , 0 )$] and #math.equation(block: false, alt: "open parenthesis − 3 , 0 close parenthesis .")[$( "−" 3 , 0 ) .$]], )) Now, we will find the #emph[x]-intercepts of the parabola with equation #math.equation(block: false, alt: "y equals minus x squared plus 4 x plus 3")[$y = − x^(2) + 4 x + 3$]. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays the quadratic equation y = -x^2 + 4x + 3, written in a clear, sans-serif font against a plain white background.], alt: "The image displays the quadratic equation y = -x^2 + 4x + 3, written in a clear, sans-serif font against a plain white background.", caption: none)]), [Let #math.equation(block: false, alt: "y equals 0")[$y = 0$].], [#figure(figph[A mathematical equation on a white background reads '0 = -x^2 + 4x + 3'.], alt: "A mathematical equation on a white background reads '0 = -x^2 + 4x + 3'.", caption: none)], [This quadratic does not factor, so we use the Quadratic Formula.], [#figure(figph[The quadratic formula: x equals negative b, plus or minus the square root of b squared minus 4ac, all divided by 2a.], alt: "The quadratic formula: x equals negative b, plus or minus the square root of b squared minus 4ac, all divided by 2a.", caption: none)], [#math.equation(block: false, alt: "a equals −1")[$a = −1$], #math.equation(block: false, alt: "b equals 4")[$b = 4$], #math.equation(block: false, alt: "c equals 3")[$c = 3$]], [#figure(figph[A mathematical equation showing the quadratic formula applied to solve for X, with specific numerical values substituted for a, b, and c.], alt: "A mathematical equation showing the quadratic formula applied to solve for X, with specific numerical values substituted for a, b, and c.", caption: none)], [Simplify.], [#figure(figph[A mathematical equation where X is equal to a fraction. The numerator of the fraction is -4 plus or minus the square root of 28, and the denominator is -2.], alt: "A mathematical equation where X is equal to a fraction. The numerator of the fraction is -4 plus or minus the square root of 28, and the denominator is -2.", caption: none) #linebreak() #figure(figph[A mathematical equation for X, showing X equals a fraction where the numerator is -4 plus or minus 2 times the square root of 7, and the denominator is -2.], alt: "A mathematical equation for X, showing X equals a fraction where the numerator is -4 plus or minus 2 times the square root of 7, and the denominator is -2.", caption: none) #linebreak() #figure(figph[A mathematical equation showing x equals the fraction where the numerator is -2 times the quantity of 2 plus or minus the square root of 7, and the denominator is -2.], alt: "A mathematical equation showing x equals the fraction where the numerator is -2 times the quantity of 2 plus or minus the square root of 7, and the denominator is -2.", caption: none) #figure(figph[A mathematical equation shows 'x = 2 ×1 ×7' on a white background. This indicates two possible solutions for x: 2 plus the square root of 7, and 2 minus the square root of 7. It's likely the result of a quadratic formula calculation.], alt: "A mathematical equation shows 'x = 2 ×1 ×7' on a white background. This indicates two possible solutions for x: 2 plus the square root of 7, and 2 minus the square root of 7. It's likely the result of a quadratic formula calculation.", caption: none)], [], [The #emph[x] intercepts are #math.equation(block: false, alt: "open parenthesis 2 plus the square root of 7 , 0 close parenthesis")[$( 2 + sqrt(7) , 0 )$] and #math.equation(block: false, alt: "open parenthesis 2 minus the square root of 7 , 0 close parenthesis")[$( 2 − sqrt(7) , 0 )$].], )) We will use the decimal approximations of the x-intercepts, so that we can locate these points on the graph. #math.equation(block: true, alt: "open parenthesis 2 plus the square root of 7 , 0 close parenthesis approximately equals open parenthesis 4.6 , 0 close parenthesis, open parenthesis 2 minus the square root of 7 , 0 close parenthesis approximately equals open parenthesis −0.6 , 0 close parenthesis")[$( 2 + sqrt(7) , 0 ) ≈ ( 4.6 , 0 ) & & & ( 2 − sqrt(7) , 0 ) ≈ ( −0.6 , 0 )$]Do these results agree with our graphs? #figure(figph[This figure shows an two graphs side by side. The graph on the left side shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The vertex is at the point (-2, -1). Three points are plotted on the curve at (-3, 0), (-1, 0), and (0, 3). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -2. Below the graph is the equation of the graph, y equals x squared plus 4 x plus 3. Below that is the statement “y-intercept (0, 3)”. Below that is the statement “x-intercepts (-1, 0) and (-3, 0)”. The graph on the right side shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The vertex is at the point (2, 7). Three points are plotted on the curve at (-0.6, 0), (4.6, 0), and (0, 3). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 2. Below the graph is the equation of the graph, y equals negative x squared plus 4 x plus 3. Below that is the statement “y-intercept (0, 3)”. Below that is the statement “x-intercepts (2 plus square root of 7, 0) is approximately equal to (4.6, 0) and (2 minus square root of 7, 0) is approximately equal to (-0.6, 0).”], alt: "This figure shows an two graphs side by side. The graph on the left side shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The vertex is at the point (-2, -1). Three points are plotted on the curve at (-3, 0), (-1, 0), and (0, 3). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -2. Below the graph is the equation of the graph, y equals x squared plus 4 x plus 3. Below that is the statement “y-intercept (0, 3)”. Below that is the statement “x-intercepts (-1, 0) and (-3, 0)”. The graph on the right side shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The vertex is at the point (2, 7). Three points are plotted on the curve at (-0.6, 0), (4.6, 0), and (0, 3). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 2. Below the graph is the equation of the graph, y equals negative x squared plus 4 x plus 3. Below that is the statement “y-intercept (0, 3)”. Below that is the statement “x-intercepts (2 plus square root of 7, 0) is approximately equal to (4.6, 0) and (2 minus square root of 7, 0) is approximately equal to (-0.6, 0).”", caption: none) #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Find the intercepts of a parabola.] To find the intercepts of a parabola with equation #math.equation(block: false, alt: "y equals a x squared plus b x plus c")[$y = a x^(2) + b x + c$]: #math.equation(block: true, alt: "y -intercept, x -intercepts; Let x equals 0 and solve for y ., Let y equals 0 and solve for x .")[$"y" "-intercept" & & & #h(4em) "x" "-intercepts" \ "Let" #h(0.2em) x = 0 #h(0.2em) "and solve for" #h(0.2em) y . & & & #h(4em) "Let" #h(0.2em) y = 0 #h(0.2em) "and solve for" #h(0.2em) x .$] ] #examplebox("Example 4")[][ Find the intercepts of the parabola #math.equation(block: false, alt: "y equals x squared minus 2 x minus 8")[$y = x^(2) − 2 x − 8$]. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays the quadratic equation y = x^2 - 2x - 8, presented in a clear, digital font on a white background.], alt: "The image displays the quadratic equation y = x^2 - 2x - 8, presented in a clear, digital font on a white background.", caption: none)]), [To find the #emph[y]-intercept, let #math.equation(block: false, alt: "x equals 0")[$x = 0$] and solve for #emph[y].], [#figure(figph[A mathematical equation shows y = 0^2 - 2 \* 0 - 8, which simplifies to y = -8. The zeros are highlighted in red, indicating a substitution or specific point of interest in the calculation.], alt: "A mathematical equation shows y = 0^2 - 2 * 0 - 8, which simplifies to y = -8. The zeros are highlighted in red, indicating a substitution or specific point of interest in the calculation.", caption: none)], [], [When #math.equation(block: false, alt: "x equals 0")[$x = 0$], then #math.equation(block: false, alt: "y equals −8")[$y = −8$]. #linebreak() The #emph[y]-intercept is the point #math.equation(block: false, alt: "open parenthesis 0 , −8 close parenthesis")[$( 0 , −8 )$].], [], [#figure(figph[The image shows a mathematical equation on a white background. The equation is 'y = x^2 - 2x - 8' written in black text.], alt: "The image shows a mathematical equation on a white background. The equation is 'y = x^2 - 2x - 8' written in black text.", caption: none)], [To find the #emph[x]-intercept, let #math.equation(block: false, alt: "y equals 0")[$y = 0$] and solve for #emph[x].], [#figure(figph[A white background displays a clear, black mathematical equation: 0 = x^2 - 2x - 8, representing a quadratic equation in standard form.], alt: "A white background displays a clear, black mathematical equation: 0 = x^2 - 2x - 8, representing a quadratic equation in standard form.", caption: none)], [Solve by factoring.], [#figure(figph[A mathematical equation shows 0 equals the product of two binomials: (x - 4) and (x + 2). This is a factored form of a quadratic equation.], alt: "A mathematical equation shows 0 equals the product of two binomials: (x - 4) and (x + 2). This is a factored form of a quadratic equation.", caption: none)], [], [#figure(figph[Solution steps for two linear equations: 0 = x - 4 is solved to x = 4, and 0 = x + 2 is solved to x = -2.], alt: "Solution steps for two linear equations: 0 = x - 4 is solved to x = 4, and 0 = x + 2 is solved to x = -2.", caption: none)], )) When #math.equation(block: false, alt: "y equals 0")[$y = 0$], then #math.equation(block: false, alt: "x equals 4 or x equals −2")[$x = 4 #h(0.2em) "or" #h(0.2em) x = −2$]. The #emph[x]-intercepts are the points #math.equation(block: false, alt: "open parenthesis 4 , 0 close parenthesis")[$( 4 , 0 )$] and #math.equation(block: false, alt: "open parenthesis −2 , 0 close parenthesis")[$( −2 , 0 )$]. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the intercepts of the parabola #math.equation(block: false, alt: "y equals x squared plus 2 x minus 8 .")[$y = x^(2) + 2 x − 8 .$] #solutionbox[ #math.equation(block: true, alt: "y : open parenthesis 0 , −8 close parenthesis ; x : open parenthesis −4 , 0 close parenthesis , open parenthesis 2 , 0 close parenthesis")[$y ":" #h(0.2em) ( 0 , −8 ) ; x ":" #h(0.2em) ( −4 , 0 ) , ( 2 , 0 )$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the intercepts of the parabola #math.equation(block: false, alt: "y equals x squared minus 4 x minus 12 .")[$y = x^(2) − 4 x − 12 .$] #solutionbox[ #math.equation(block: true, alt: "y : open parenthesis 0 , −12 close parenthesis ; x : open parenthesis 6 , 0 close parenthesis , open parenthesis −2 , 0 close parenthesis")[$y ":" #h(0.2em) ( 0 , −12 ) ; x ":" #h(0.2em) ( 6 , 0 ) , ( −2 , 0 )$] ] ] In this chapter, we have been solving quadratic equations of the form #math.equation(block: false, alt: "a x squared plus b x plus c equals 0")[$a x^(2) + b x + c = 0$]. We solved for #math.equation(block: false, alt: "x")[$x$] and the results were the solutions to the equation. We are now looking at quadratic equations in two variables of the form #math.equation(block: false, alt: "y equals a x squared plus b x plus c")[$y = a x^(2) + b x + c$]. The graphs of these equations are parabolas. The #emph[x]-intercepts of the parabolas occur where #math.equation(block: false, alt: "y equals 0")[$y = 0$]. For example: #math.equation(block: true, alt: "Quadratic equation, Quadratic equation in two variables; y equals x squared minus 2 x minus 15; x squared minus 2 x minus 15, equals, 0; open parenthesis x minus 5 close parenthesis open parenthesis x plus 3 close parenthesis, equals, 0, let y equals 0 0 equals x squared minus 2 x minus 15; 0 equals open parenthesis x minus 5 close parenthesis open parenthesis x plus 3 close parenthesis; x minus 5, equals, 0, x plus 3, equals, 0; x, equals, 5, x, equals, −3, x minus 5, equals, 0, x plus 3, equals, 0; x, equals, 5, x, equals, −3; open parenthesis 5 , 0 close parenthesis and open parenthesis −3 , 0 close parenthesis; x -intercepts")[$"Quadratic equation" & & & #h(4em) "Quadratic equation in two variables" \ & & & #h(4em) y = x^(2) − 2 x − 15 \ x^(2) − 2 x − 15 & = & 0 \ ( x − 5 ) ( x + 3 ) & = & 0 & & & #h(4em) "let" #h(0.2em) y = 0 #h(1em) 0 = x^(2) − 2 x − 15 \ 0 = ( x − 5 ) ( x + 3 ) \ x − 5 & = & 0 & & & x + 3 & = & 0 \ x & = & 5 & & & x & = & −3 & & & #h(4em) x − 5 & = & 0 & & & x + 3 & = & 0 \ x & = & 5 & & & x & = & −3 \ & & & #h(4em) ( 5 , 0 ) #h(0.2em) "and" #h(0.2em) ( −3 , 0 ) \ & & & #h(4em) x "-intercepts"$]The solutions of the quadratic equation are the #math.equation(block: false, alt: "x")[$x$] values of the #emph[x]-intercepts. Earlier, we saw that quadratic equations have 2, 1, or 0 solutions. The graphs below show examples of parabolas for these three cases. Since the solutions of the equations give the #emph[x]-intercepts of the graphs, the number of #emph[x]-intercepts is the same as the number of solutions. Previously, we used the discriminant to determine the number of solutions of a quadratic equation of the form #math.equation(block: false, alt: "a x squared plus b x plus c equals 0")[$a x^(2) + b x + c = 0$]. Now, we can use the discriminant to tell us how many #emph[x]-intercepts there are on the graph. #figure(figph[This figure shows three graphs side by side. The leftmost graph shows an upward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola is in the lower right quadrant. Below the graph is the inequality b squared minus 4 a c greater than 0. Below that is the statement “Two solutions”. Below that is the statement “ Two x-intercepts”. The middle graph shows an downward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola is on the x-axis. Below the graph is the equation b squared minus 4 a c equals 0. Below that is the statement “One solution”. Below that is the statement “ One x-intercept”. The rightmost graph shows an upward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola is in the upper left quadrant. Below the graph is the inequality b squared minus 4 a c less than 0. Below that is the statement “No real solutions”. Below that is the statement “ No x-intercept”.], alt: "This figure shows three graphs side by side. The leftmost graph shows an upward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola is in the lower right quadrant. Below the graph is the inequality b squared minus 4 a c greater than 0. Below that is the statement “Two solutions”. Below that is the statement “ Two x-intercepts”. The middle graph shows an downward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola is on the x-axis. Below the graph is the equation b squared minus 4 a c equals 0. Below that is the statement “One solution”. Below that is the statement “ One x-intercept”. The rightmost graph shows an upward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola is in the upper left quadrant. Below the graph is the inequality b squared minus 4 a c less than 0. Below that is the statement “No real solutions”. Below that is the statement “ No x-intercept”.", caption: none) Before you start solving the quadratic equation to find the values of the #emph[x]-intercepts, you may want to evaluate the discriminant so you know how many solutions to expect. #examplebox("Example 5")[][ Find the intercepts of the parabola #math.equation(block: false, alt: "y equals 5 x squared plus x plus 4")[$y = 5 x^(2) + x + 4$]. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays the quadratic equation y = 5x^2 + x + 4, presented in a clear, standard mathematical format.], alt: "The image displays the quadratic equation y = 5x^2 + x + 4, presented in a clear, standard mathematical format.", caption: none)]), [To find the #emph[y]-intercept, let #math.equation(block: false, alt: "x equals 0")[$x = 0$] and solve for #emph[y].], [#figure(figph[A mathematical equation y = 5 \* 0^2 + 0 + 4, with zeros highlighted in red, demonstrating a calculation involving zero.], alt: "A mathematical equation y = 5 * 0^2 + 0 + 4, with zeros highlighted in red, demonstrating a calculation involving zero.", caption: none) #linebreak() #figure(figph[The equation y = 4 is displayed in black text against a white background.], alt: "The equation y = 4 is displayed in black text against a white background.", caption: none) #linebreak() When #math.equation(block: false, alt: "x equals 0")[$x = 0$], then #math.equation(block: false, alt: "y equals 4")[$y = 4$]. #linebreak() The #emph[y]-intercept is the point #math.equation(block: false, alt: "open parenthesis 0 , 4 close parenthesis")[$( 0 , 4 )$].], [], [#figure(figph[A mathematical equation, 'y = 5x^2 + x + 4,' is displayed against a white background.], alt: "A mathematical equation, 'y = 5x^2 + x + 4,' is displayed against a white background.", caption: none)], [To find the #emph[x]-intercept, let #math.equation(block: false, alt: "y equals 0")[$y = 0$] and solve for #emph[x].], [#figure(figph[The quadratic equation 0 = 5x^2 + x + 4 is displayed.], alt: "The quadratic equation 0 = 5x^2 + x + 4 is displayed.", caption: none)], [Find the value of the discriminant to predict the number of solutions and so #emph[x]-intercepts.], [#math.equation(block: false, alt: "b squared, minus, 4 a c; 1 squared, minus, 4 times 5 times 4; 1, minus, 80; −79")[$b^(2) & − & 4 a c \ 1^(2) & − & 4 ⋅ 5 ⋅ 4 \ 1 & − & 80 \ −79 #h(0.8em)$]], [Since the value of the discriminant is negative, there is no real solution to the equation.], [There are no #emph[x]-intercepts.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the intercepts of the parabola #math.equation(block: false, alt: "y equals 3 x squared plus 4 x plus 4 .")[$y = 3 x^(2) + 4 x + 4 .$] #solutionbox[ #math.equation(block: true, alt: "y : open parenthesis 0 , 4 close parenthesis ; x : none")[$y ":" #h(0.2em) ( 0 , 4 ) ; x ":" #h(0.2em) "none"$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the intercepts of the parabola #math.equation(block: false, alt: "y equals x squared minus 4 x minus 5 .")[$y = x^(2) − 4 x − 5 .$] #solutionbox[ #math.equation(block: true, alt: "y : open parenthesis 0 , −5 close parenthesis ; x : open parenthesis 5 , 0 close parenthesis open parenthesis −1 , 0 close parenthesis")[$y ":" #h(0.2em) ( 0 , −5 ) ; x ":" #h(0.2em) ( 5 , 0 ) #h(0.2em) ( −1 , 0 )$] ] ] #examplebox("Example 6")[][ Find the intercepts of the parabola #math.equation(block: false, alt: "y equals 4 x squared minus 12 x plus 9")[$y = 4 x^(2) − 12 x + 9$]. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image shows the quadratic equation y = 4x^2 - 12x + 9.], alt: "The image shows the quadratic equation y = 4x^2 - 12x + 9.", caption: none)]), [To find the #emph[y]-intercept, let #math.equation(block: false, alt: "x equals 0")[$x = 0$] and solve for #emph[y].], [#figure(figph[The image shows the mathematical equation y = 4 \* 0^2 - 12 \* 0 + 9, where the zeros are highlighted in red.], alt: "The image shows the mathematical equation y = 4 * 0^2 - 12 * 0 + 9, where the zeros are highlighted in red.", caption: none) #linebreak() #figure(figph[The equation y = 9 is displayed in black text against a plain white background, indicating a constant value for y in a mathematical or scientific context.], alt: "The equation y = 9 is displayed in black text against a plain white background, indicating a constant value for y in a mathematical or scientific context.", caption: none)], [], [When #math.equation(block: false, alt: "x equals 0")[$x = 0$], then #math.equation(block: false, alt: "y equals 9")[$y = 9$]. #linebreak() The #emph[y]-intercept is the point #math.equation(block: false, alt: "open parenthesis 0 , 9 close parenthesis")[$( 0 , 9 )$].], [], [#figure(figph[The image shows the quadratic equation y = 4x^2 - 12x + 9 in white text on a black background.], alt: "The image shows the quadratic equation y = 4x^2 - 12x + 9 in white text on a black background.", caption: none)], [To find the #emph[x]-intercept, let #math.equation(block: false, alt: "y equals 0")[$y = 0$] and solve for #emph[x].], [#figure(figph[A quadratic equation is displayed, reading 0 = 4x^2 - 12x + 9.], alt: "A quadratic equation is displayed, reading 0 = 4x^2 - 12x + 9.", caption: none)], [Find the value of the discriminant to predict the number of solutions and so #emph[x]-intercepts.], [#math.equation(block: false, alt: "b squared, minus, 4 a c; 2 squared, minus, 4 times 4 times 9; 144, minus, 144; 0")[$b^(2) & − & 4 a c \ 2^(2) & − & 4 ⋅ 4 ⋅ 9 \ 144 & − & 144 \ 0 #h(1em)$]], [], [Since the value of the discriminant is 0, there is only one real solution to the equation. Therefore, there is only one #emph[x]-intercept.], [Solve the equation by factoring the perfect square trinomial.], [#figure(figph[The equation 0 = (2x - 3)^2 is displayed on a white background, representing a quadratic equation in vertex form.], alt: "The equation 0 = (2x - 3)^2 is displayed on a white background, representing a quadratic equation in vertex form.", caption: none)], [Use the Zero Product Property.], [#figure(figph[A mathematical equation, 0 = 2x - 3, is displayed.], alt: "A mathematical equation, 0 = 2x - 3, is displayed.", caption: none)], [Solve for #emph[x].], [#figure(figph[The equation 3 = 2x is displayed on a white background, representing a basic algebraic problem.], alt: "The equation 3 = 2x is displayed on a white background, representing a basic algebraic problem.", caption: none) #linebreak() #figure(figph[The image shows the mathematical equation three over two equals x (3/2 = x), presented in a simple, clear format on a white background.], alt: "The image shows the mathematical equation three over two equals x (3/2 = x), presented in a simple, clear format on a white background.", caption: none)], [], [When #math.equation(block: false, alt: "y equals 0")[$y = 0$], then #math.equation(block: false, alt: "the fraction 3 over 2 equals x .")[$frac(3, 2) = x .$]], [], [The #emph[x]-intercept is the point #math.equation(block: false, alt: "open parenthesis the fraction 3 over 2 , 0 close parenthesis .")[$( frac(3, 2) , 0 ) .$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the intercepts of the parabola #math.equation(block: false, alt: "y equals − x squared minus 12 x minus 36 .")[$y = "−" x^(2) − 12 x − 36 .$] #solutionbox[ #math.equation(block: true, alt: "y : open parenthesis 0 , −36 close parenthesis ; x : open parenthesis −6 , 0 close parenthesis")[$y ":" #h(0.2em) ( 0 , −36 ) ; x ":" #h(0.2em) ( −6 , 0 )$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the intercepts of the parabola #math.equation(block: false, alt: "y equals 9 x squared plus 12 x plus 4 .")[$y = 9 x^(2) + 12 x + 4 .$] #solutionbox[ #math.equation(block: true, alt: "y : open parenthesis 0 , 4 close parenthesis ; x : open parenthesis minus the fraction 2 over 3 , 0 close parenthesis")[$y ":" #h(0.2em) ( 0 , 4 ) ; x ":" #h(0.2em) ( − frac(2, 3) , 0 )$] ] ] === Graph Quadratic Equations in Two Variables Now, we have all the pieces we need in order to graph a quadratic equation in two variables. We just need to put them together. In the next example, we will see how to do this. #examplebox("Example 7")[How To Graph a Quadratic Equation in Two Variables][ Graph #math.equation(block: false, alt: "y equals x squared minus 6 x plus 8")[$y = x^(2) − 6 x + 8$]. #solutionbox[ #figure(figph[The image shows the steps to graph the quadratic equation y equals x squared minus 6 x plus 8. Step 1 is to write the quadratic equation with y on one side. This equation has y on one side already. The value of a is one, the value of b is -6 and the value of c is 8.], alt: "The image shows the steps to graph the quadratic equation y equals x squared minus 6 x plus 8. Step 1 is to write the quadratic equation with y on one side. This equation has y on one side already. The value of a is one, the value of b is -6 and the value of c is 8.", caption: none) #figure(figph[Step 2 is to determine whether the parabola opens upward or downward. Since a is positive, the parabola opens upward.], alt: "Step 2 is to determine whether the parabola opens upward or downward. Since a is positive, the parabola opens upward.", caption: none) #figure(figph[Step 3 is to find the axis of symmetry. The axis of symmetry is the line x equals negative b divided by the quantity 2 a. Plugging in the values of b and a the formula becomes x equals negative -6 divided by the quantity 2 times 1 which simplifies to x equals 3. The axis of symmetry is the line x equals 3.], alt: "Step 3 is to find the axis of symmetry. The axis of symmetry is the line x equals negative b divided by the quantity 2 a. Plugging in the values of b and a the formula becomes x equals negative -6 divided by the quantity 2 times 1 which simplifies to x equals 3. The axis of symmetry is the line x equals 3.", caption: none) #figure(figph[Step 4 is to find the vertex. The vertex is on the axis of symmetry. Substitute x equals 3 into the equation and solve for y. The equation is y equals x squared minus 6 x plus 8. Replacing x with 3 it becomes y equals 3 squared minus 6 times 3 plus 8 which simplifies to y equals -1. The vertex is (3, -1).], alt: "Step 4 is to find the vertex. The vertex is on the axis of symmetry. Substitute x equals 3 into the equation and solve for y. The equation is y equals x squared minus 6 x plus 8. Replacing x with 3 it becomes y equals 3 squared minus 6 times 3 plus 8 which simplifies to y equals -1. The vertex is (3, -1).", caption: none) #figure(figph[Step 5 is to find the y-intercept and find the point symmetric to the y-intercept across the axis of symmetry. We substitute x equals 0 into the equation. The equation is y equals x squared minus 6 x plus 8. Replacing x with 0 it becomes y equals 0 squared minus 6 times 0 plus 8 which simplifies to y equals 8. The y-intercept is (0, 8). We use the axis of symmetry to find a point symmetric to the y-intercept. The y-intercept is 3 units left of the axis of symmetry, x equals 3. A point 3 units to the right of the axis of symmetry has x equals 6. The point symmetric to the y-intercept is (6, 8).], alt: "Step 5 is to find the y-intercept and find the point symmetric to the y-intercept across the axis of symmetry. We substitute x equals 0 into the equation. The equation is y equals x squared minus 6 x plus 8. Replacing x with 0 it becomes y equals 0 squared minus 6 times 0 plus 8 which simplifies to y equals 8. The y-intercept is (0, 8). We use the axis of symmetry to find a point symmetric to the y-intercept. The y-intercept is 3 units left of the axis of symmetry, x equals 3. A point 3 units to the right of the axis of symmetry has x equals 6. The point symmetric to the y-intercept is (6, 8).", caption: none) #figure(figph[Step 6 is to find the x-intercepts. We substitute y equals 0 into the equation. The equation becomes 0 equals x squared minus 6 x plus 8. We can solve this quadratic equation by factoring to get 0 equals the quantity x minus 2 times the quantity x minus 4. Solve each equation to get x equals 2 and x equals 4. The x-intercepts are (2, 0) and (4, 0).], alt: "Step 6 is to find the x-intercepts. We substitute y equals 0 into the equation. The equation becomes 0 equals x squared minus 6 x plus 8. We can solve this quadratic equation by factoring to get 0 equals the quantity x minus 2 times the quantity x minus 4. Solve each equation to get x equals 2 and x equals 4. The x-intercepts are (2, 0) and (4, 0).", caption: none) #figure(figph[Step 7 is to graph the parabola. We graph the vertex, intercepts, and the point symmetric to the y-intercept. We connect these five points to sketch the parabola. The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -2 to 10. The y-axis of the plane runs from -3 to 10. The vertex is at the point (3, -1). Four points are plotted on the curve at (0, 8), (6, 8), (2, 0) and (4, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 3.], alt: "Step 7 is to graph the parabola. We graph the vertex, intercepts, and the point symmetric to the y-intercept. We connect these five points to sketch the parabola. The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -2 to 10. The y-axis of the plane runs from -3 to 10. The vertex is at the point (3, -1). Four points are plotted on the curve at (0, 8), (6, 8), (2, 0) and (4, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 3.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the parabola #math.equation(block: false, alt: "y equals x squared plus 2 x minus 8 .")[$y = x^(2) + 2 x − 8 .$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , −8 close parenthesis")[$y ":" #h(0.2em) ( 0 , −8 )$]; #math.equation(block: false, alt: "x : open parenthesis 2 , 0 close parenthesis , open parenthesis −4 , 0 close parenthesis")[$x ":" #h(0.2em) ( 2 , 0 ) , ( −4 , 0 )$]; #linebreak() axis: #math.equation(block: false, alt: "x equals −1")[$x = −1$]; vertex: #math.equation(block: false, alt: "open parenthesis −1 , −9 close parenthesis")[$( −1 , −9 )$]; #linebreak() #figure(figph[The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The vertex is at the point (-1, -9). Three points are plotted on the curve at (0, -8), (2, 0) and (-4, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -1.], alt: "The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The vertex is at the point (-1, -9). Three points are plotted on the curve at (0, -8), (2, 0) and (-4, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -1.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the parabola #math.equation(block: false, alt: "y equals x squared minus 8 x plus 12 .")[$y = x^(2) − 8 x + 12 .$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , 12 close parenthesis ; x : open parenthesis 2 , 0 close parenthesis , open parenthesis 6 , 0 close parenthesis ;")[$y ":" #h(0.2em) ( 0 , 12 ) ; #h(0.2em) x ":" #h(0.2em) ( 2 , 0 ) , ( 6 , 0 ) ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals 4 ; vertex: open parenthesis 4 , −4 close parenthesis")[$x = 4 ; #h(0.2em) "vertex:" #h(0.2em) ( 4 , −4 )$]; #linebreak() #figure(figph[The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The vertex is at the point (4, -4). Three points are plotted on the curve at (0, 12), (2, 0) and (6, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 4.], alt: "The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The vertex is at the point (4, -4). Three points are plotted on the curve at (0, 12), (2, 0) and (6, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 4.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Graph a quadratic equation in two variables.] + Write the quadratic equation with #math.equation(block: false, alt: "y")[$y$] on one side. + Determine whether the parabola opens upward or downward. + Find the axis of symmetry. + Find the vertex. + Find the #emph[y]-intercept. Find the point symmetric to the #emph[y]-intercept across the axis of symmetry. + Find the #emph[x]-intercepts. + Graph the parabola. ] We were able to find the #emph[x]-intercepts in the last example by factoring. We find the #emph[x]-intercepts in the next example by factoring, too. #examplebox("Example 8")[][ Graph #math.equation(block: false, alt: "y equals − x squared plus 6 x minus 9")[$y = "−" x^(2) + 6 x − 9$]. #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [The equation #emph[y] has on one side.], [#figure(figph[A general quadratic equation y = ax^2 + bx + c is shown above a specific example, y = -x^2 + 6x - 9.], alt: "A general quadratic equation y = ax^2 + bx + c is shown above a specific example, y = -x^2 + 6x - 9.", caption: none)], [], [Since #emph[a] is #math.equation(block: false, alt: "minus 1")[$− 1$], the parabola opens downward. #linebreak() #linebreak() To find the axis of symmetry, find #math.equation(block: false, alt: "x equals minus the fraction b over 2 a")[$x = − frac(b, 2 a)$].], [#figure(figph[A red arc connects two downward-pointing arrows, suggesting a path or relationship between two descending points or outcomes.], alt: "A red arc connects two downward-pointing arrows, suggesting a path or relationship between two descending points or outcomes.", caption: none)], [#figure(figph[The formula x = -b / (2a) is displayed, which is used to find the x-coordinate of the vertex of a parabola for a quadratic equation.], alt: "The formula x = -b / (2a) is displayed, which is used to find the x-coordinate of the vertex of a parabola for a quadratic equation.", caption: none) #linebreak() #figure(figph[A mathematical equation shows X equals negative 6 divided by the product of 2 and negative 1, written as X = -6 / (2(-1)).], alt: "A mathematical equation shows X equals negative 6 divided by the product of 2 and negative 1, written as X = -6 / (2(-1)).", caption: none) #linebreak() #figure(figph[The mathematical equation 'x=3' is displayed in black text on a plain white background.], alt: "The mathematical equation 'x=3' is displayed in black text on a plain white background.", caption: none) #linebreak() The axis of symmetry is #math.equation(block: false, alt: "x equals 3 .")[$x = 3 .$] The vertex is on the line #math.equation(block: false, alt: "x equals 3 .")[$x = 3 .$] #linebreak() #figure(figph[A coordinate plane shows a vertical dashed red line at x=3, intersecting the x-axis at 3 and extending indefinitely through points such as (3,4), (3,0), and (3,-10).], alt: "A coordinate plane shows a vertical dashed red line at x=3, intersecting the x-axis at 3 and extending indefinitely through points such as (3,4), (3,0), and (3,-10).", caption: none)], [Find #emph[y] when #math.equation(block: false, alt: "x equals 3 .")[$x = 3 .$]], [#figure(figph[A mathematical equation is displayed on a white background: y = -x^2 + 6x - 9. This represents a quadratic function, likely a parabola opening downwards.], alt: "A mathematical equation is displayed on a white background: y = -x^2 + 6x - 9. This represents a quadratic function, likely a parabola opening downwards.", caption: none) #linebreak() #figure(figph[Mathematical expression y = -3 squared + 6 \* 3 - 9; the number 3 is highlighted in red to indicate its use as a variable for substitution.], alt: "Mathematical expression y = -3 squared + 6 * 3 - 9; the number 3 is highlighted in red to indicate its use as a variable for substitution.", caption: none) #linebreak() #figure(figph[A mathematical equation is displayed, showing y = -9 + 18 - 9, where y is equal to negative nine plus eighteen minus nine.], alt: "A mathematical equation is displayed, showing y = -9 + 18 - 9, where y is equal to negative nine plus eighteen minus nine.", caption: none) #linebreak() #figure(figph[The mathematical equation 'y = 0' is displayed in black font against a white background.], alt: "The mathematical equation 'y = 0' is displayed in black font against a white background.", caption: none) #linebreak() The vertex is #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis .")[$( 3 , 0 ) .$] #linebreak() #figure(figph[A graph displays a vertical dashed line representing the equation x=3, intersecting the x-axis at (3,0).], alt: "A graph displays a vertical dashed line representing the equation x=3, intersecting the x-axis at (3,0).", caption: none)], [], [The #emph[y]-intercept occurs when #math.equation(block: false, alt: "x equals 0 .")[$x = 0 .$] #linebreak() Substitute #math.equation(block: false, alt: "x equals 0 .")[$x = 0 .$] #linebreak() Simplify. #linebreak() #linebreak() The point #math.equation(block: false, alt: "open parenthesis 0 , −9 close parenthesis")[$( 0 , −9 )$] is three units to the left of the line of symmetry. #linebreak() The point three units to the right of the line of symmetry is #math.equation(block: false, alt: "open parenthesis 6 , −9 close parenthesis .")[$( 6 , −9 ) .$] #linebreak() Point symmetric to the #emph[y-]intercept is #math.equation(block: false, alt: "open parenthesis 6 , −9 close parenthesis")[$( 6 , −9 )$]], [#figure(figph[A mathematical equation is displayed, showing y = -x^2 + 6x - 9. This represents a quadratic function, likely to be graphed as a parabola opening downwards.], alt: "A mathematical equation is displayed, showing y = -x^2 + 6x - 9. This represents a quadratic function, likely to be graphed as a parabola opening downwards.", caption: none) #linebreak() #figure(figph[The mathematical equation y = -0^2 + 6 '.' 0 - 9 is displayed, where y is calculated by substituting 0 into a quadratic expression.], alt: "The mathematical equation y = -0^2 + 6 '.' 0 - 9 is displayed, where y is calculated by substituting 0 into a quadratic expression.", caption: none) #linebreak() #figure(figph[The image displays the simple algebraic equation 'y = -9' in black text on a plain white background.], alt: "The image displays the simple algebraic equation 'y = -9' in black text on a plain white background.", caption: none) #linebreak() The #emph[y]-intercept is #math.equation(block: false, alt: "open parenthesis 0 , −9 close parenthesis .")[$( 0 , −9 ) .$] #linebreak() #figure(figph[This graph illustrates a vertical dashed line at x=3 and three distinct points on a Cartesian coordinate system. One point is on the line, and two others are horizontally aligned at y=-9.], alt: "This graph illustrates a vertical dashed line at x=3 and three distinct points on a Cartesian coordinate system. One point is on the line, and two others are horizontally aligned at y=-9.", caption: none)], [], [The #emph[x]-intercept occurs when #math.equation(block: false, alt: "y equals 0 .")[$y = 0 .$]], [#figure(figph[A mathematical equation is displayed, showing y = -x^2 + 6x - 9, which represents a quadratic function and could be used for graphing a parabola opening downwards.], alt: "A mathematical equation is displayed, showing y = -x^2 + 6x - 9, which represents a quadratic function and could be used for graphing a parabola opening downwards.", caption: none)], [], [Substitute #math.equation(block: false, alt: "y equals 0 .")[$y = 0 .$]], [#figure(figph[A quadratic equation displayed as 0 = -x^2 + 6x - 9.], alt: "A quadratic equation displayed as 0 = -x^2 + 6x - 9.", caption: none)], [], [Factor the GCF.], [#figure(figph[A mathematical equation shows '0 = -(x^2 - 6x + 9)' in black text on a white background, representing a quadratic expression set equal to zero with a negative sign outside the parentheses.], alt: "A mathematical equation shows '0 = -(x^2 - 6x + 9)' in black text on a white background, representing a quadratic expression set equal to zero with a negative sign outside the parentheses.", caption: none)], [], [Factor the trinomial.], [#figure(figph[The image shows the mathematical equation 0 = -(x - 3)^2 in a clear, digital font against a white background.], alt: "The image shows the mathematical equation 0 = -(x - 3)^2 in a clear, digital font against a white background.", caption: none)], [], [Solve for #emph[x].], [#figure(figph[The image displays a simple algebraic equation, 'X = 3', set against a plain white background.], alt: "The image displays a simple algebraic equation, 'X = 3', set against a plain white background.", caption: none)], [], [Connect the points to graph the parabola.], [#figure(figph[A graph displays a downward-opening parabola with its vertex at (3, 0). A dashed red line indicates the axis of symmetry at x = 3. Two points on the parabola are (1, -9) and (5, -9).], alt: "A graph displays a downward-opening parabola with its vertex at (3, 0). A dashed red line indicates the axis of symmetry at x = 3. Two points on the parabola are (1, -9) and (5, -9).", caption: none)], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the parabola #math.equation(block: false, alt: "y equals −3 x squared plus 12 x minus 12 .")[$y = −3 x^(2) + 12 x − 12 .$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , −12 close parenthesis ; x : open parenthesis 2 , 0 close parenthesis ;")[$y ":" #h(0.2em) ( 0 , −12 ) ; #h(0.2em) x ":" #h(0.2em) ( 2 , 0 ) ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals 2 ; vertex: open parenthesis 2 , 0 close parenthesis")[$x = 2 ; #h(0.2em) "vertex:" ( 2 , 0 )$]; #linebreak() #figure(figph[The graph shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -1 to 10. The vertex is at the point (2, 0). One other point is plotted on the curve at (0, -12). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 2.], alt: "The graph shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -1 to 10. The vertex is at the point (2, 0). One other point is plotted on the curve at (0, -12). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 2.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the parabola #math.equation(block: false, alt: "y equals 25 x squared plus 10 x plus 1 .")[$y = 25 x^(2) + 10 x + 1 .$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , 1 close parenthesis ; x : open parenthesis minus the fraction 1 over 5 , 0 close parenthesis ;")[$y ":" #h(0.2em) ( 0 , 1 ) ; #h(0.2em) x ":" #h(0.2em) ( − frac(1, 5) , 0 ) ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals minus the fraction 1 over 5 ; vertex: open parenthesis minus the fraction 1 over 5 , 0 close parenthesis")[$x = − frac(1, 5) ; #h(0.2em) "vertex:" #h(0.2em) ( − frac(1, 5) , 0 )$]; #linebreak() #figure(figph[The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -5 to 5. The y-axis of the plane runs from -5 to 10. The vertex is at the point (-1 fifth, 0). One other point is plotted on the curve at (0, 1). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -1 fifth.], alt: "The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -5 to 5. The y-axis of the plane runs from -5 to 10. The vertex is at the point (-1 fifth, 0). One other point is plotted on the curve at (0, 1). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -1 fifth.", caption: none) ] ] For the graph of #math.equation(block: false, alt: "y equals minus x squared plus 6 x minus 9")[$y = − x^(2) + 6 x − 9$], the vertex and the #emph[x]-intercept were the same point. Remember how the discriminant determines the number of solutions of a quadratic equation? The discriminant of the equation #math.equation(block: false, alt: "0 equals − x squared plus 6 x minus 9")[$0 = "−" x^(2) + 6 x − 9$] is 0, so there is only one solution. That means there is only one #emph[x]-intercept, and it is the vertex of the parabola. How many #emph[x]-intercepts would you expect to see on the graph of #math.equation(block: false, alt: "y equals x squared plus 4 x plus 5")[$y = x^(2) + 4 x + 5$]? #examplebox("Example 9")[][ Graph #math.equation(block: false, alt: "y equals x squared plus 4 x plus 5")[$y = x^(2) + 4 x + 5$]. #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, [The equation has #emph[y] on one side.], [#figure(figph[The image presents the general quadratic equation y = ax^2 + bx + c, alongside a specific numerical example: y = x^2 + 4x + 5.], alt: "The image presents the general quadratic equation y = ax^2 + bx + c, alongside a specific numerical example: y = x^2 + 4x + 5.", caption: none)], [], [Since a is 1, the parabola opens upward.], [#figure(figph[Two red arrows point upwards, creating a V-shape against a white background, suggesting an upward trend or divergence.], alt: "Two red arrows point upwards, creating a V-shape against a white background, suggesting an upward trend or divergence.", caption: none)], [], [To find the axis of symmetry, find #math.equation(block: false, alt: "x equals minus the fraction b over 2 a .")[$x = − frac(b, 2 a) .$]], [#figure(figph[The mathematical formula x = -b / 2a, used to find the x-coordinate of the vertex of a parabola, is displayed on a white background.], alt: "The mathematical formula x = -b / 2a, used to find the x-coordinate of the vertex of a parabola, is displayed on a white background.", caption: none) #linebreak() #figure(figph[A mathematical expression shows X equals negative 4 divided by the product of 2 and 1.], alt: "A mathematical expression shows X equals negative 4 divided by the product of 2 and 1.", caption: none) #linebreak() #figure(figph[A close-up view of a simple mathematical equation, X = -2, displayed clearly on a white background.], alt: "A close-up view of a simple mathematical equation, X = -2, displayed clearly on a white background.", caption: none) #linebreak() The axis of symmetry is #math.equation(block: false, alt: "x equals −2 .")[$x = −2 .$] #linebreak() #figure(figph[A Cartesian coordinate plane shows a vertical dashed red line at x = -2. The x and y axes both range from -6 to 6, with grid lines every unit.], alt: "A Cartesian coordinate plane shows a vertical dashed red line at x = -2. The x and y axes both range from -6 to 6, with grid lines every unit.", caption: none)], [], [The vertex is on the line #math.equation(block: false, alt: "x equals −2 .")[$x = −2 .$]], [], [], [Find #emph[y] when #math.equation(block: false, alt: "x equals −2 .")[$x = −2 .$]], [#figure(figph[The image displays the quadratic equation y = x^2 + 4x + 5 in black text against a white background.], alt: "The image displays the quadratic equation y = x^2 + 4x + 5 in black text against a white background.", caption: none) #linebreak() #figure(figph[A mathematical equation is displayed, showing y = (-2)^2 + 4 \* (-2) + 5. This expression involves squaring a negative number, multiplying, and adding constants, likely for calculation or evaluation.], alt: "A mathematical equation is displayed, showing y = (-2)^2 + 4 * (-2) + 5. This expression involves squaring a negative number, multiplying, and adding constants, likely for calculation or evaluation.", caption: none) #linebreak() #figure(figph[A mathematical equation is displayed, showing 'y = 4 - 8 + 5' in a clear, dark font against a white background.], alt: "A mathematical equation is displayed, showing 'y = 4 - 8 + 5' in a clear, dark font against a white background.", caption: none) #linebreak() #figure(figph[The image displays the mathematical equation 'y = 1' in black text against a plain white background. The equation indicates that the variable 'y' is equal to the number '1'.], alt: "The image displays the mathematical equation 'y = 1' in black text against a plain white background. The equation indicates that the variable 'y' is equal to the number '1'.", caption: none) #linebreak() The vertex is #math.equation(block: false, alt: "open parenthesis −2 , 1 close parenthesis .")[$( −2 , 1 ) .$] #linebreak() #figure(figph[A coordinate plane displays a vertical dashed red line at x = -2. A light blue point is plotted on this line at coordinates (-2, 1).], alt: "A coordinate plane displays a vertical dashed red line at x = -2. A light blue point is plotted on this line at coordinates (-2, 1).", caption: none)], [], [The #emph[y]-intercept occurs when #math.equation(block: false, alt: "x equals 0 .")[$x = 0 .$] #linebreak() Substitute #math.equation(block: false, alt: "x equals 0 .")[$x = 0 .$] #linebreak() Simplify. #linebreak() The point #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$] is two units to the right of the line of symmetry. #linebreak() The point two units to the left of the line of symmetry is #math.equation(block: false, alt: "open parenthesis −4 , 5 close parenthesis .")[$( −4 , 5 ) .$]], [#figure(figph[The image displays the quadratic equation y = x^2 + 4x + 5 in black text on a white background.], alt: "The image displays the quadratic equation y = x^2 + 4x + 5 in black text on a white background.", caption: none) #linebreak() #figure(figph[The mathematical equation y = (0)^2 + 4(0) + 5 is shown, with the number '0' highlighted in red within parentheses.], alt: "The mathematical equation y = (0)^2 + 4(0) + 5 is shown, with the number '0' highlighted in red within parentheses.", caption: none) #linebreak() #figure(figph[The mathematical equation 'y = 5' is displayed on a white background, indicating a constant value for the variable y.], alt: "The mathematical equation 'y = 5' is displayed on a white background, indicating a constant value for the variable y.", caption: none) #linebreak() The #emph[y]-intercept is #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis .")[$( 0 , 5 ) .$] #linebreak() #figure(figph[A coordinate plane shows three blue points: (-4, 5), (-2, 1), and (0, 5). A vertical dashed red line is drawn at x = -2.], alt: "A coordinate plane shows three blue points: (-4, 5), (-2, 1), and (0, 5). A vertical dashed red line is drawn at x = -2.", caption: none) #linebreak() Point symmetric to the #emph[y-] intercept is #math.equation(block: false, alt: "open parenthesis −4 , 5 close parenthesis")[$( −4 , 5 )$].], [], [The #emph[x]- intercept occurs when #math.equation(block: false, alt: "y equals 0 .")[$y = 0 .$]], [], [], [Substitute #math.equation(block: false, alt: "y equals 0 .")[$y = 0 .$] #linebreak() Test the discriminant.], [#figure(figph[The image displays the quadratic equation y = x^2 + 4x + 5, written in a clear, standard mathematical format.], alt: "The image displays the quadratic equation y = x^2 + 4x + 5, written in a clear, standard mathematical format.", caption: none) #linebreak() #figure(figph[A mathematical quadratic equation, 0 equals x squared plus 4x plus 5, is displayed on a white background, representing a standard algebraic problem.], alt: "A mathematical quadratic equation, 0 equals x squared plus 4x plus 5, is displayed on a white background, representing a standard algebraic problem.", caption: none)], [], [], [], [#math.equation(block: false, alt: "b squared minus 4 a c")[$b^(2) − 4 a c$] #linebreak() #math.equation(block: false, alt: "4 squared minus 4 times 1 5")[$4^(2) − 4 ⋅ 1 5$] #linebreak() #math.equation(block: false, alt: "16 minus 20")[$16 − 20$] #linebreak() #math.equation(block: false, alt: "−4")[$#h(1em) −4$]], [Since the value of the discriminant is negative, there is no solution and so no #emph[x-] intercept. #linebreak() Connect the points to graph the parabola. You may want to choose two more points for greater accuracy.], [#figure(figph[A parabola with its vertex at (-2, 1) and axis of symmetry x = -2 is plotted on a coordinate plane. The parabola opens upwards, passing through points (-4, 5) and (0, 5).], alt: "A parabola with its vertex at (-2, 1) and axis of symmetry x = -2 is plotted on a coordinate plane. The parabola opens upwards, passing through points (-4, 5) and (0, 5).", caption: none)], [], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the parabola #math.equation(block: false, alt: "y equals 2 x squared minus 6 x plus 5 .")[$y = 2 x^(2) − 6 x + 5 .$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , 5 close parenthesis ; x : none ;")[$y ":" #h(0.2em) ( 0 , 5 ) ; #h(0.2em) x ":" #h(0.2em) "none" ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals the fraction 3 over 2 ; vertex: open parenthesis the fraction 3 over 2 , the fraction 1 over 2 close parenthesis")[$x = frac(3, 2) ; #h(0.2em) "vertex:" #h(0.2em) ( frac(3, 2) , frac(1, 2) )$]; #linebreak() #figure(figph[The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -5 to 5. The y-axis of the plane runs from -5 to 10. The vertex is at the point (3 halves, 1 half). One other point is plotted on the curve at (0, 5). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 3 halves.], alt: "The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -5 to 5. The y-axis of the plane runs from -5 to 10. The vertex is at the point (3 halves, 1 half). One other point is plotted on the curve at (0, 5). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 3 halves.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the parabola #math.equation(block: false, alt: "y equals −2 x squared minus 1 .")[$y = −2 x^(2) − 1 .$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , −1 close parenthesis ; x : none ;")[$y ":" #h(0.2em) ( 0 , −1 ) ; x ":" #h(0.2em) "none" ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals 0 ; vertex: open parenthesis 0 , −1 close parenthesis")[$x = 0 ; #h(0.2em) "vertex:" #h(0.2em) ( 0 , −1 )$]; #linebreak() #figure(figph[The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The vertex is at the point (0, -1). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 0.], alt: "The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The vertex is at the point (0, -1). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 0.", caption: none) ] ] Finding the #emph[y]-intercept by substituting #math.equation(block: false, alt: "x equals 0")[$x = 0$] into the equation is easy, isn’t it? But we needed to use the Quadratic Formula to find the #emph[x]-intercepts in . We will use the Quadratic Formula again in the next example. #examplebox("Example 10")[][ Graph #math.equation(block: false, alt: "y equals 2 x squared minus 4 x minus 3")[$y = 2 x^(2) − 4 x − 3$]. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays the general form of a quadratic equation, y = ax^2 + bx + c, in red, followed by a specific example, y = 2x^2 - 4x - 3, in black, illustrating the application of the formula.], alt: "The image displays the general form of a quadratic equation, y = ax^2 + bx + c, in red, followed by a specific example, y = 2x^2 - 4x - 3, in black, illustrating the application of the formula.", caption: none)]), [The equation #emph[y] has one side. #linebreak() Since #emph[a] is 2, the parabola opens upward.], [#figure(figph[Two sleek red arrows curve upward to form a sharp 'V' shape, highlighting an ascent or choice against a minimalist white background. The design is simple yet conveys direction.], alt: "Two sleek red arrows curve upward to form a sharp 'V' shape, highlighting an ascent or choice against a minimalist white background. The design is simple yet conveys direction.", caption: none)], [To find the axis of symmetry, find #math.equation(block: false, alt: "x equals minus the fraction b over 2 a")[$x = − frac(b, 2 a)$].], [#figure(figph[The mathematical formula x = -b / 2a is displayed on a white background. This formula represents the x-coordinate of the vertex of a parabola, which is derived from the quadratic formula.], alt: "The mathematical formula x = -b / 2a is displayed on a white background. This formula represents the x-coordinate of the vertex of a parabola, which is derived from the quadratic formula.", caption: none) #linebreak() #figure(figph[A mathematical equation on a white background, displaying X = -4 / (2 \* 2).], alt: "A mathematical equation on a white background, displaying X = -4 / (2 * 2).", caption: none) #linebreak() #figure(figph[The mathematical equation 'x = 1' is displayed in black text on a plain white background.], alt: "The mathematical equation 'x = 1' is displayed in black text on a plain white background.", caption: none) #linebreak() The axis of symmetry is #math.equation(block: false, alt: "x equals 1")[$x = 1$].], [The vertex on the line #math.equation(block: false, alt: "x equals 1 .")[$x = 1 .$]], [#figure(figph[The image displays the quadratic equation y = 2x^2 - 4x - 3 on a white background. The equation is rendered in black text.], alt: "The image displays the quadratic equation y = 2x^2 - 4x - 3 on a white background. The equation is rendered in black text.", caption: none)], [Find #emph[y] when #math.equation(block: false, alt: "x equals 1")[$x = 1$].], [#figure(figph[A mathematical equation is displayed, showing y = 2(1)^2 - 4 \* (1) - 3, likely as a step in solving a quadratic function by substituting x=1.], alt: "A mathematical equation is displayed, showing y = 2(1)^2 - 4 * (1) - 3, likely as a step in solving a quadratic function by substituting x=1.", caption: none) #linebreak() #figure(figph[A mathematical equation is displayed on a white background: y = 2 - 4 - 3.], alt: "A mathematical equation is displayed on a white background: y = 2 - 4 - 3.", caption: none) #linebreak() #figure(figph[The equation y = -5 is displayed against a white background.], alt: "The equation y = -5 is displayed against a white background.", caption: none) #linebreak() The vertex is #math.equation(block: false, alt: "open parenthesis 1 , − 5 close parenthesis")[$( 1 , "−" 5 )$].], [The #emph[y]-intercept occurs when #math.equation(block: false, alt: "x equals 0 .")[$x = 0 .$]], [#figure(figph[The image displays the quadratic equation y = 2x^2 - 4x - 3 written in a clear, standard mathematical notation against a plain white background.], alt: "The image displays the quadratic equation y = 2x^2 - 4x - 3 written in a clear, standard mathematical notation against a plain white background.", caption: none)], [Substitute #math.equation(block: false, alt: "x equals 0 .")[$x = 0 .$]], [#figure(figph[The equation y = 2 \* 0^2 - 4 \* 0 - 3, demonstrating the substitution of 0 into a quadratic expression.], alt: "The equation y = 2 * 0^2 - 4 * 0 - 3, demonstrating the substitution of 0 into a quadratic expression.", caption: none)], [Simplify.], [#figure(figph[The image displays the mathematical equation 'y = -3' written in plain text on a white background, representing a horizontal line in a coordinate system.], alt: "The image displays the mathematical equation 'y = -3' written in plain text on a white background, representing a horizontal line in a coordinate system.", caption: none) #linebreak() The #emph[y-]intercept is #math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis")[$( 0 , −3 )$].], [The point #math.equation(block: false, alt: "open parenthesis 0 , −3 close parenthesis")[$( 0 , −3 )$] is one unit to the left of the line of symmetry. #linebreak() The point one unit to the right of the line of symmetry is #math.equation(block: false, alt: "open parenthesis 2 , −3 close parenthesis")[$( 2 , −3 )$]], [Point symmetric to the #emph[y-]intercept is #math.equation(block: false, alt: "open parenthesis 2 , −3 close parenthesis .")[$( 2 , −3 ) .$]], [The #emph[x]-intercept occurs when #math.equation(block: false, alt: "y equals 0")[$y = 0$].], [#figure(figph[A mathematical equation is displayed on a white background: y = 2x^2 - 4x - 3. The equation is rendered in a black sans-serif font.], alt: "A mathematical equation is displayed on a white background: y = 2x^2 - 4x - 3. The equation is rendered in a black sans-serif font.", caption: none)], [Substitute #math.equation(block: false, alt: "y equals 0")[$y = 0$].], [#figure(figph[A quadratic equation, 0 = 2x² - 4x - 3, is displayed in black text against a white background.], alt: "A quadratic equation, 0 = 2x² - 4x - 3, is displayed in black text against a white background.", caption: none)], [Use the Quadratic Formula.], [#figure(figph[The quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / 2a, used to find the roots of a quadratic equation.], alt: "The quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / 2a, used to find the roots of a quadratic equation.", caption: none)], [Substitute in the values of #emph[a, b, c.]], [#figure(figph[The quadratic formula applied, showing the substitution of specific numerical values for a, b, and c to calculate the roots of a quadratic equation.], alt: "The quadratic formula applied, showing the substitution of specific numerical values for a, b, and c to calculate the roots of a quadratic equation.", caption: none)], [Simplify.], [#figure(figph[A mathematical equation shows x equals a fraction where the numerator is 4 plus or minus the square root of (16 plus 24), all divided by 4.], alt: "A mathematical equation shows x equals a fraction where the numerator is 4 plus or minus the square root of (16 plus 24), all divided by 4.", caption: none)], [Simplify inside the radical.], [#figure(figph[A mathematical equation showing x equals 4 plus or minus the square root of 40, all divided by 4.], alt: "A mathematical equation showing x equals 4 plus or minus the square root of 40, all divided by 4.", caption: none)], [Simplify the radical.], [#figure(figph[A mathematical equation is displayed, showing x equals the fraction 4 plus or minus 2 times the square root of 10, all divided by 4.], alt: "A mathematical equation is displayed, showing x equals the fraction 4 plus or minus 2 times the square root of 10, all divided by 4.", caption: none)], [Factor the GCF.], [#figure(figph[A mathematical equation shows x equals the fraction with a numerator of two times the quantity of two plus or minus the square root of ten, and a denominator of four.], alt: "A mathematical equation shows x equals the fraction with a numerator of two times the quantity of two plus or minus the square root of ten, and a denominator of four.", caption: none)], [Remove common factors.], [#figure(figph[A mathematical equation displays x equals a fraction where the numerator is 2 plus or minus the square root of 10, and the denominator is 2.], alt: "A mathematical equation displays x equals a fraction where the numerator is 2 plus or minus the square root of 10, and the denominator is 2.", caption: none)], [Write as two equations.], [#figure(figph[The two solutions for x from a quadratic equation, expressed as (2 + sqrt(10))/2 and (2 - sqrt(10))/2.], alt: "The two solutions for x from a quadratic equation, expressed as (2 + sqrt(10))/2 and (2 - sqrt(10))/2.", caption: none)], [Approximate the values.], [#figure(figph[Two approximate solutions for 'x' are displayed: x is approximately 2.5 and x is approximately -0.6, representing numerical results often found in mathematical problems.], alt: "Two approximate solutions for 'x' are displayed: x is approximately 2.5 and x is approximately -0.6, representing numerical results often found in mathematical problems.", caption: none)], [], [The approximate values of the #emph[x-]intercepts are #math.equation(block: false, alt: "open parenthesis 2.5 , 0 close parenthesis")[$( 2.5 , 0 )$] and #math.equation(block: false, alt: "open parenthesis −0.6 , 0 close parenthesis")[$( −0.6 , 0 )$].], [Graph the parabola using the points found.], [#figure(figph[A graph showing an upward-opening parabola with its vertex at (1, -5). The dashed vertical line x=1 represents its axis of symmetry. Two additional points, (0, -3) and (2, -3), are marked.], alt: "A graph showing an upward-opening parabola with its vertex at (1, -5). The dashed vertical line x=1 represents its axis of symmetry. Two additional points, (0, -3) and (2, -3), are marked.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the parabola #math.equation(block: false, alt: "y equals 5 x squared plus 10 x plus 3 .")[$y = 5 x^(2) + 10 x + 3 .$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , 3 close parenthesis ; x : open parenthesis −1.6 , 0 close parenthesis , open parenthesis −0.4 , 0 close parenthesis ;")[$y ":" #h(0.2em) ( 0 , 3 ) ; #h(0.2em) x ":" #h(0.2em) ( −1.6 , 0 ) , ( −0.4 , 0 ) ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals −1 ; vertex: open parenthesis −1 , −2 close parenthesis")[$x = −1 ; #h(0.2em) "vertex:" #h(0.2em) ( −1 , −2 )$]; #linebreak() #figure(figph[The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -5 to 5. The y-axis of the plane runs from -5 to 5. The vertex is at the point (-1,-2). Three other points are plotted on the curve at (0, 3), (-1.6, 0), (-0.4, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -1.], alt: "The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -5 to 5. The y-axis of the plane runs from -5 to 5. The vertex is at the point (-1,-2). Three other points are plotted on the curve at (0, 3), (-1.6, 0), (-0.4, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -1.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Graph the parabola #math.equation(block: false, alt: "y equals −3 x squared minus 6 x plus 5 .")[$y = −3 x^(2) − 6 x + 5 .$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , 5 close parenthesis ; x : open parenthesis 0.6 , 0 close parenthesis , open parenthesis −2.6 , 0 close parenthesis ;")[$y ":" #h(0.2em) ( 0 , 5 ) ; #h(0.2em) x ":" #h(0.2em) ( 0.6 , 0 ) , ( −2.6 , 0 ) ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals −1 ; vertex: open parenthesis −1 , 8 close parenthesis")[$x = −1 ; #h(0.2em) "vertex:" #h(0.2em) ( −1 , 8 )$]; #linebreak() #figure(figph[The graph shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The vertex is at the point (-1, 8). Three other points are plotted on the curve at (0, 5), (0.6, 0) and (-2.6, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -1.], alt: "The graph shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The vertex is at the point (-1, 8). Three other points are plotted on the curve at (0, 5), (0.6, 0) and (-2.6, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -1.", caption: none) ] ] === Solve Maximum and Minimum Applications Knowing that the #strong[vertex] of a parabola is the lowest or highest point of the parabola gives us an easy way to determine the minimum or maximum value of a quadratic equation. The #emph[y]-coordinate of the vertex is the minimum #emph[y]-value of a parabola that opens upward. It is the maximum #emph[y]-value of a parabola that opens downward. #figure(figph[This figure shows two graphs side by side. The left graph shows an downward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola is in the upper right quadrant. The vertex is labeled “maximum”. The right graph shows an upward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola is in the lower right quadrant. The vertex is labeled “minimum”.], alt: "This figure shows two graphs side by side. The left graph shows an downward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola is in the upper right quadrant. The vertex is labeled “maximum”. The right graph shows an upward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola is in the lower right quadrant. The vertex is labeled “minimum”.", caption: none) #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Minimum or Maximum Values of a Quadratic Equation] The #strong[#emph[y]-coordinate of the vertex] of the graph of a quadratic equation is the - minimum value of the quadratic equation if the parabola opens upward. - maximum value of the quadratic equation if the parabola opens downward. ] #examplebox("Example 11")[][ Find the minimum value of the quadratic equation #math.equation(block: false, alt: "y equals x squared plus 2 x minus 8")[$y = x^(2) + 2 x − 8$]. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[A mathematical equation is displayed: y = x^2 + 2x - 8. The equation is rendered in a clear, digital font against a plain white background, focusing entirely on the algebraic expression.], alt: "A mathematical equation is displayed: y = x^2 + 2x - 8. The equation is rendered in a clear, digital font against a plain white background, focusing entirely on the algebraic expression.", caption: none)]), [Since #emph[a] is positive, the parabola opens upward.], [], [The quadratic equation has a minimum.], [], [Find the axis of symmetry.], [#figure(figph[A mathematical equation displays 'x = -b/2a' in black text on a white background. This formula is used to find the x-coordinate of the vertex of a parabola in a quadratic equation.], alt: "A mathematical equation displays 'x = -b/2a' in black text on a white background. This formula is used to find the x-coordinate of the vertex of a parabola in a quadratic equation.", caption: none) #linebreak() #figure(figph[A mathematical equation shows x equals negative 2 divided by the product of 2 and 1, written as x = -2/(2\*1).], alt: "A mathematical equation shows x equals negative 2 divided by the product of 2 and 1, written as x = -2/(2*1).", caption: none) #linebreak() #figure(figph[The image displays a mathematical equation in plain black text against a white background, which states 'X = -1'.], alt: "The image displays a mathematical equation in plain black text against a white background, which states 'X = -1'.", caption: none) #linebreak() The axis of symmetry is #math.equation(block: false, alt: "x equals −1")[$x = −1$].], [The vertex is on the line #math.equation(block: false, alt: "x equals −1 .")[$x = −1 .$]], [#figure(figph[The algebraic equation y = x^2 + 2x - 8 is shown.], alt: "The algebraic equation y = x^2 + 2x - 8 is shown.", caption: none)], [Find #emph[y] when #math.equation(block: false, alt: "x equals −1 .")[$x = −1 .$]], [#figure(figph[The image shows the equation y = (-1)^2 + 2(-1) - 8, which is an evaluation of a quadratic function at x = -1.], alt: "The image shows the equation y = (-1)^2 + 2(-1) - 8, which is an evaluation of a quadratic function at x = -1.", caption: none) #linebreak() #figure(figph[A mathematical equation showing y equals one minus two minus eight (y=1-2-8) on a white background.], alt: "A mathematical equation showing y equals one minus two minus eight (y=1-2-8) on a white background.", caption: none) #linebreak() #figure(figph[The image displays the mathematical equation 'y = -9' in simple black text against a plain white background, indicating a horizontal line on a Cartesian plane.], alt: "The image displays the mathematical equation 'y = -9' in simple black text against a plain white background, indicating a horizontal line on a Cartesian plane.", caption: none) #linebreak() The vertex is #math.equation(block: false, alt: "open parenthesis −1 , −9 close parenthesis")[$( −1 , −9 )$].], [Since the parabola has a minimum, the #emph[y-]coordinate of the vertex is the minimum #emph[y-]value of the quadratic equation.], [], [The minimum value of the quadratic is #math.equation(block: false, alt: "−9")[$−9$] and it occurs when #math.equation(block: false, alt: "x equals −1")[$x = −1$].], [], [Show the graph to verify the result.], [#figure(figph[A parabola opening upwards is plotted on a coordinate plane, with its vertex at (-1, -9). A dashed red vertical line at x=-1 represents the axis of symmetry.], alt: "A parabola opening upwards is plotted on a coordinate plane, with its vertex at (-1, -9). A dashed red vertical line at x=-1 represents the axis of symmetry.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the maximum or minimum value of the quadratic equation #math.equation(block: false, alt: "y equals x squared minus 8 x plus 12")[$y = x^(2) − 8 x + 12$]. #solutionbox[ The minimum value is #math.equation(block: false, alt: "−4")[$−4$] when #math.equation(block: false, alt: "x equals 4")[$x = 4$]. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the maximum or minimum value of the quadratic equation #math.equation(block: false, alt: "y equals −4 x squared plus 16 x minus 11")[$y = −4 x^(2) + 16 x − 11$]. #solutionbox[ The maximum value is 5 when #math.equation(block: false, alt: "x equals 2")[$x = 2$]. ] ] We have used the formula #math.equation(block: true, alt: "h equals −16 t squared plus v sub 0 t plus h sub 0")[$h = −16 t^(2) + v_(0) t + h_(0)$]to calculate the height in feet, #math.equation(block: false, alt: "h")[$h$], of an object shot upwards into the air with initial velocity, #math.equation(block: false, alt: "v sub 0")[$v_(0)$], after #math.equation(block: false, alt: "t")[$t$] seconds. This formula is a quadratic equation in the variable #math.equation(block: false, alt: "t")[$t$], so its graph is a parabola. By solving for the coordinates of the vertex, we can find how long it will take the object to reach its maximum height. Then, we can calculate the maximum height. #examplebox("Example 12")[][ The quadratic equation #math.equation(block: false, alt: "h equals −16 t squared plus v sub 0 t plus h sub 0")[$h = −16 t^(2) + v_(0) t + h_(0)$] models the height of a volleyball hit straight upwards with velocity 176 feet per second from a height of 4 feet. + ⓐ How many seconds will it take the volleyball to reach its maximum height? + ⓑ Find the maximum height of the volleyball. #solutionbox[ #math.equation(block: true, alt: "h equals −16 t squared plus 176 t plus 4")[$h = −16 t^(2) + 176 t + 4$] Since a is negative, the parabola opens downward. The quadratic equation has a maximum. + ⓐ #linebreak() #math.equation(block: false, alt: "Find the axis of symmetry., t equals minus the fraction b over 2 a; t equals minus the fraction 176 over 2 open parenthesis −16 close parenthesis; t equals 5.5; The axis of symmetry is t equals 5.5 .; The vertex is on the line t equals 5.5 ., The maximum occurs when t equals 5.5 seconds.")[$"Find the axis of symmetry." & & & #h(4em) t = − frac(b, 2 a) \ t = − frac(176, 2 ( −16 )) \ t = 5.5 \ & & & #h(4em) "The axis of symmetry is" #h(0.2em) t = 5.5 . \ "The vertex is on the line" #h(0.2em) t = 5.5 . & & & #h(4em) "The maximum occurs when" #h(0.2em) t = 5.5 #h(0.2em) "seconds."$] + ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([Find #emph[h] when #math.equation(block: false, alt: "t equals 5.5")[$t = 5.5$].], [#figure(figph[A mathematical equation is displayed, which reads: h = -16t^2 + 176t + 4. This is a quadratic equation, often used to model projectile motion or other parabolic phenomena.], alt: "A mathematical equation is displayed, which reads: h = -16t^2 + 176t + 4. This is a quadratic equation, often used to model projectile motion or other parabolic phenomena.", caption: none) #linebreak() #figure(figph[A mathematical equation is displayed, showing h = -16(5.5)^2 + 176 \* (5.5) + 4, where the numbers 5.5 are highlighted in red.], alt: "A mathematical equation is displayed, showing h = -16(5.5)^2 + 176 * (5.5) + 4, where the numbers 5.5 are highlighted in red.", caption: none)]), [Use a calculator to simplify.], [#figure(figph[The image displays the text 'h = 488' in a simple, clear font on a white background. It appears to be a variable assignment or a measurement value.], alt: "The image displays the text 'h = 488' in a simple, clear font on a white background. It appears to be a variable assignment or a measurement value.", caption: none)], [], [The vertex is #math.equation(block: false, alt: "open parenthesis 5.5 , 488 close parenthesis")[$( 5.5 , 488 )$].], [Since the parabola has a maximum, the #emph[h-]coordinate of the vertex is the maximum #emph[y]-value of the quadratic equation.], [The maximum value of the quadratic is 488 feet and it occurs when #math.equation(block: false, alt: "t equals 5.5")[$t = 5.5$] seconds.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The quadratic equation #math.equation(block: false, alt: "h equals −16 t squared plus 128 t plus 32")[$h = −16 t^(2) + 128 t + 32$] is used to find the height of a stone thrown upward from a height of 32 feet at a rate of 128 ft/sec. How long will it take for the stone to reach its maximum height? What is the maximum height? Round answers to the nearest tenth. #solutionbox[ It will take 4 seconds to reach the maximum height of 288 feet. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ A toy rocket shot upward from the ground at a rate of 208 ft/sec has the quadratic equation of #math.equation(block: false, alt: "h equals −16 t squared plus 208 t")[$h = −16 t^(2) + 208 t$]. When will the rocket reach its maximum height? What will be the maximum height? Round answers to the nearest tenth. #solutionbox[ It will take 6.5 seconds to reach the maximum height of 676 feet. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Access these online resources for additional instruction and practice graphing quadratic equations: - #link("https://openstax.org/l/25Graphquad1")[Graphing Quadratic Functions] - #link("https://openstax.org/l/25Graphquad2")[How do you graph a quadratic function?] - #link("https://openstax.org/l/25Graphquad3")[Graphing Quadratic Equations] ] === Key Concepts - #strong[The graph of every quadratic equation is a parabola.] - #strong[Parabola Orientation] For the quadratic equation #math.equation(block: false, alt: "y equals a x squared plus b x plus c")[$y = a x^(2) + b x + c$], if - #math.equation(block: false, alt: "a greater than 0")[$a > 0$], the parabola opens upward. - #math.equation(block: false, alt: "a less than 0")[$a < 0$], the parabola opens downward. - #strong[Axis of Symmetry and Vertex of a Parabola] For a parabola with equation #math.equation(block: false, alt: "y equals a x squared plus b x plus c")[$y = a x^(2) + b x + c$]: - The axis of symmetry of a parabola is the line #math.equation(block: false, alt: "x equals minus the fraction b over 2 a")[$x = − frac(b, 2 a)$]. - The vertex is on the axis of symmetry, so its #emph[x]-coordinate is #math.equation(block: false, alt: "minus the fraction b over 2 a")[$− frac(b, 2 a)$]. - To find the #emph[y]-coordinate of the vertex we substitute #math.equation(block: false, alt: "x equals minus the fraction b over 2 a")[$x = − frac(b, 2 a)$] into the quadratic equation. - #strong[Find the Intercepts of a Parabola] To find the intercepts of a parabola with equation #math.equation(block: false, alt: "y equals a x squared plus b x plus c")[$y = a x^(2) + b x + c$]: #linebreak() #math.equation(block: false, alt: "y -intercept, x -intercepts; Let x equals 0 and solve for y ., Let y equals 0 and solve for x .")[$"y" "-intercept" & & & "x" "-intercepts" \ "Let" #h(0.2em) x = 0 #h(0.2em) "and solve for" #h(0.2em) y . & & & "Let" #h(0.2em) y = 0 #h(0.2em) "and solve for" #h(0.2em) x .$] - #strong[To Graph a Quadratic Equation in Two Variables] + Write the quadratic equation with #math.equation(block: false, alt: "y")[$y$] on one side. + Determine whether the parabola opens upward or downward. + Find the axis of symmetry. + Find the vertex. + Find the #emph[y]-intercept. Find the point symmetric to the #emph[y]-intercept across the axis of symmetry. + Find the #emph[x]-intercepts. + Graph the parabola. - #strong[Minimum or Maximum Values of a Quadratic Equation] - The #emph[y]-#strong[coordinate of the vertex] of the graph of a quadratic equation is the - #strong[minimum] value of the quadratic equation if the parabola opens upward. - #strong[maximum] value of the quadratic equation if the parabola opens downward. === Section Exercises ==== Practice Makes Perfect #strong[Recognize the Graph of a Quadratic Equation in Two Variables] In the following exercises, graph: #math.equation(block: true, alt: "y equals x squared plus 3")[$y = x^(2) + 3$] #solutionbox[ #figure(figph[This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has a vertex at (0, 3) and goes through the point (1, 4).], alt: "This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has a vertex at (0, 3) and goes through the point (1, 4).", caption: none) ] #math.equation(block: true, alt: "y equals − x squared plus 1")[$y = "−" x^(2) + 1$] In the following exercises, determine if the parabola opens up or down. #math.equation(block: true, alt: "y equals −2 x squared minus 6 x minus 7")[$y = −2 x^(2) − 6 x − 7$] #solutionbox[ down ] #math.equation(block: true, alt: "y equals 6 x squared plus 2 x plus 3")[$y = 6 x^(2) + 2 x + 3$] #math.equation(block: true, alt: "y equals 4 x squared plus x minus 4")[$y = 4 x^(2) + x − 4$] #solutionbox[ up ] #math.equation(block: true, alt: "y equals −9 x squared minus 24 x minus 16")[$y = −9 x^(2) − 24 x − 16$] #strong[Find the Axis of Symmetry and Vertex of a Parabola] In the following exercises, find ⓐ the axis of symmetry and ⓑ the vertex. #math.equation(block: true, alt: "y equals x squared plus 8 x minus 1")[$y = x^(2) + 8 x − 1$] #solutionbox[ ⓐ #math.equation(block: false, alt: "x equals −4")[$x = −4$] ⓑ #math.equation(block: false, alt: "open parenthesis −4 , −17 close parenthesis")[$( −4 , −17 )$] ] #math.equation(block: true, alt: "y equals x squared plus 10 x plus 25")[$y = x^(2) + 10 x + 25$] #math.equation(block: true, alt: "y equals − x squared plus 2 x plus 5")[$y = "−" x^(2) + 2 x + 5$] #solutionbox[ ⓐ #math.equation(block: false, alt: "x equals 1")[$x = 1$] ⓑ #math.equation(block: false, alt: "open parenthesis 1 , 6 close parenthesis")[$( 1 , 6 )$] ] #math.equation(block: true, alt: "y equals −2 x squared minus 8 x minus 3")[$y = −2 x^(2) − 8 x − 3$] #strong[Find the Intercepts of a Parabola] In the following exercises, find the #emph[x]- and #emph[y]-intercepts. #math.equation(block: true, alt: "y equals x squared plus 7 x plus 6")[$y = x^(2) + 7 x + 6$] #solutionbox[ #math.equation(block: true, alt: "y : open parenthesis 0 , 6 close parenthesis ; x : open parenthesis −1 , 0 close parenthesis , open parenthesis −6 , 0 close parenthesis")[$y ":" #h(0.2em) ( 0 , 6 ) ; #h(0.2em) x ":" #h(0.2em) ( −1 , 0 ) , ( −6 , 0 )$] ] #math.equation(block: true, alt: "y equals x squared plus 10 x minus 11")[$y = x^(2) + 10 x − 11$] #math.equation(block: true, alt: "y equals − x squared plus 8 x minus 19")[$y = "−" x^(2) + 8 x − 19$] #solutionbox[ #math.equation(block: true, alt: "y : open parenthesis 0 , minus 19 close parenthesis ; x : none")[$y ":" #h(0.2em) ( 0 , − 19 ) ; #h(0.2em) x ":" #h(0.2em) "none"$] ] #math.equation(block: true, alt: "y equals x squared plus 6 x plus 13")[$y = x^(2) + 6 x + 13$] #math.equation(block: true, alt: "y equals 4 x squared minus 20 x plus 25")[$y = 4 x^(2) − 20 x + 25$] #solutionbox[ #math.equation(block: true, alt: "y : open parenthesis 0 , 25 close parenthesis ; x : open parenthesis the fraction 5 over 2 , 0 close parenthesis")[$y ":" #h(0.2em) ( 0 , 25 ) ; #h(0.2em) x ":" #h(0.2em) ( frac(5, 2) , 0 )$] ] #math.equation(block: true, alt: "y equals − x squared minus 14 x minus 49")[$y = "−" x^(2) − 14 x − 49$] #strong[Graph Quadratic Equations in Two Variables] In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry. #math.equation(block: true, alt: "y equals x squared plus 6 x plus 5")[$y = x^(2) + 6 x + 5$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , 5 close parenthesis ; x : open parenthesis −1 , 0 close parenthesis , open parenthesis −5 , 0 close parenthesis ;")[$y ":" #h(0.2em) ( 0 , 5 ) ; #h(0.2em) x ":" #h(0.2em) ( −1 , 0 ) , ( −5 , 0 ) ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals −3 ; vertex : open parenthesis −3 , −4 close parenthesis")[$x = −3 ; #h(0.2em) "vertex" ":" #h(0.2em) ( −3 , −4 )$] #linebreak() #figure(figph[This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (-3, -4) and the intercepts (-5, 0), (-1, 0) and (0, 5). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -3.], alt: "This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (-3, -4) and the intercepts (-5, 0), (-1, 0) and (0, 5). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -3.", caption: none) ] #math.equation(block: true, alt: "y equals x squared plus 4 x minus 12")[$y = x^(2) + 4 x − 12$] #math.equation(block: true, alt: "y equals x squared plus 4 x plus 3")[$y = x^(2) + 4 x + 3$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , 3 close parenthesis ; x : open parenthesis −1 , 0 close parenthesis , open parenthesis −3 , 0 close parenthesis ;")[$y ":" #h(0.2em) ( 0 , 3 ) ; x ":" #h(0.2em) ( −1 , 0 ) , ( −3 , 0 ) ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals −2 ; vertex: open parenthesis −2 , −1 close parenthesis")[$x = −2 ; #h(0.2em) "vertex:" #h(0.2em) ( −2 , −1 )$] #linebreak() #figure(figph[This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (-2, -1) and the intercepts (-1, 0), (-3, 0) and (0, 3). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -2.], alt: "This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (-2, -1) and the intercepts (-1, 0), (-3, 0) and (0, 3). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -2.", caption: none) ] #math.equation(block: true, alt: "y equals x squared minus 6 x plus 8")[$y = x^(2) − 6 x + 8$] #math.equation(block: true, alt: "y equals 9 x squared plus 12 x plus 4")[$y = 9 x^(2) + 12 x + 4$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , 4 close parenthesis x : open parenthesis minus the fraction 2 over 3 , 0 close parenthesis ;")[$y ":" #h(0.2em) ( 0 , 4 ) #h(0.2em) x ":" #h(0.2em) ( − frac(2, 3) , 0 ) ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals minus the fraction 2 over 3 ; vertex: open parenthesis minus the fraction 2 over 3 , 0 close parenthesis")[$x = − frac(2, 3) ; #h(0.2em) "vertex:" #h(0.2em) ( − frac(2, 3) , 0 )$] #linebreak() #figure(figph[This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -5 to 5. The y-axis of the plane runs from -5 to 5. The parabola has points plotted at the vertex (-2 thirds, 0) and the intercept (0, 4). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -2 thirds.], alt: "This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -5 to 5. The y-axis of the plane runs from -5 to 5. The parabola has points plotted at the vertex (-2 thirds, 0) and the intercept (0, 4). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -2 thirds.", caption: none) ] #math.equation(block: true, alt: "y equals − x squared plus 8 x minus 16")[$y = "−" x^(2) + 8 x − 16$] #math.equation(block: true, alt: "y equals − x squared plus 2 x minus 7")[$y = "−" x^(2) + 2 x − 7$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , −7 close parenthesis ; x : none ;")[$y ":" #h(0.2em) ( 0 , −7 ) ; x ":" #h(0.2em) "none" ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals 1 ; vertex : open parenthesis 1 , −6 close parenthesis")[$x = 1 ; "vertex" ":" #h(0.2em) ( 1 , −6 )$] #linebreak() #figure(figph[This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -15 to 5. The parabola has points plotted at the vertex (1, -6) and the intercept (0, -7). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 1.], alt: "This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -15 to 5. The parabola has points plotted at the vertex (1, -6) and the intercept (0, -7). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 1.", caption: none) ] #math.equation(block: true, alt: "y equals 5 x squared plus 2")[$y = 5 x^(2) + 2$] #math.equation(block: true, alt: "y equals 2 x squared minus 4 x plus 1")[$y = 2 x^(2) − 4 x + 1$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , 1 close parenthesis ; x : open parenthesis 1.7 , 0 close parenthesis , open parenthesis 0.3 , 0 close parenthesis ;")[$y ":" #h(0.2em) ( 0 , 1 ) ; x ":" #h(0.2em) ( 1.7 , 0 ) , ( 0.3 , 0 ) ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals 1 ; vertex : open parenthesis 1 , −1 close parenthesis")[$x = 1 ; #h(0.2em) "vertex" ":" #h(0.2em) ( 1 , −1 )$] #linebreak() #figure(figph[This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (1, -1) and the intercepts (1.7, 0), (0.3, 0) and (0, 1). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 1.], alt: "This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (1, -1) and the intercepts (1.7, 0), (0.3, 0) and (0, 1). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 1.", caption: none) ] #math.equation(block: true, alt: "y equals 3 x squared minus 6 x minus 1")[$y = 3 x^(2) − 6 x − 1$] #math.equation(block: true, alt: "y equals 2 x squared minus 4 x plus 2")[$y = 2 x^(2) − 4 x + 2$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , 2 close parenthesis x : open parenthesis 1 , 0 close parenthesis ;")[$y ":" #h(0.2em) ( 0 , 2 ) #h(0.2em) x ":" #h(0.2em) ( 1 , 0 ) ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals 1 ; vertex: open parenthesis 1 , 0 close parenthesis")[$x = 1 ; #h(0.2em) "vertex:" #h(0.2em) ( 1 , 0 )$] #linebreak() #figure(figph[ This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (1, 0) and the intercept (0, 2). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 1.], alt: "This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (1, 0) and the intercept (0, 2). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 1.", caption: none) ] #math.equation(block: true, alt: "y equals −4 x squared minus 6 x minus 2")[$y = −4 x^(2) − 6 x − 2$] #math.equation(block: true, alt: "y equals − x squared minus 4 x plus 2")[$y = "−" x^(2) − 4 x + 2$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , 2 close parenthesis x : open parenthesis −4.4 , 0 close parenthesis , open parenthesis 0.4 , 0 close parenthesis ;")[$y ":" #h(0.2em) ( 0 , 2 ) #h(0.2em) x ":" #h(0.2em) ( −4.4 , 0 ) , ( 0.4 , 0 ) ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals −2 ; vertex: open parenthesis −2 , 6 close parenthesis")[$x = −2 ; #h(0.2em) "vertex:" #h(0.2em) ( −2 , 6 )$] #linebreak() #figure(figph[This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (-2, 6) and the intercepts (-4.4, 0), (0.4, 0) and (0, 2). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -2.], alt: "This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (-2, 6) and the intercepts (-4.4, 0), (0.4, 0) and (0, 2). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -2.", caption: none) ] #math.equation(block: true, alt: "y equals x squared plus 6 x plus 8")[$y = x^(2) + 6 x + 8$] #math.equation(block: true, alt: "y equals 5 x squared minus 10 x plus 8")[$y = 5 x^(2) − 10 x + 8$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , 8 close parenthesis ; x : none ;")[$y ":" #h(0.2em) ( 0 , 8 ) ; x ":" #h(0.2em) "none" ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals 1 ; vertex : open parenthesis 1 , 3 close parenthesis")[$x = 1 ; "vertex" ":" #h(0.2em) ( 1 , 3 )$] #linebreak() #figure(figph[This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (1, 3) and the intercept(0, 8). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 1.], alt: "This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (1, 3) and the intercept(0, 8). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 1.", caption: none) ] #math.equation(block: true, alt: "y equals −16 x squared plus 24 x minus 9")[$y = −16 x^(2) + 24 x − 9$] #math.equation(block: true, alt: "y equals 3 x squared plus 18 x plus 20")[$y = 3 x^(2) + 18 x + 20$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , 20 close parenthesis x : open parenthesis −4.5 , 0 close parenthesis , open parenthesis −1.5 , 0 close parenthesis ;")[$y ":" #h(0.2em) ( 0 , 20 ) #h(0.2em) x ":" #h(0.2em) ( −4.5 , 0 ) , ( −1.5 , 0 ) ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals −3 ; vertex: open parenthesis −3 , −7 close parenthesis")[$x = −3 ; #h(0.2em) "vertex:" #h(0.2em) ( −3 , −7 )$] #linebreak() #figure(figph[This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (-3, -7) and the intercepts (-4.5, 0) and (-1.5, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -3.], alt: "This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (-3, -7) and the intercepts (-4.5, 0) and (-1.5, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -3.", caption: none) ] #math.equation(block: true, alt: "y equals −2 x squared plus 8 x minus 10")[$y = −2 x^(2) + 8 x − 10$] #strong[Solve Maximum and Minimum Applications] In the following exercises, find the maximum or minimum value. #math.equation(block: true, alt: "y equals 2 x squared plus x minus 1")[$y = 2 x^(2) + x − 1$] #solutionbox[ The minimum value is #math.equation(block: false, alt: "minus the fraction 9 over 8")[$− frac(9, 8)$] when #math.equation(block: false, alt: "x equals minus the fraction 1 over 4")[$x = − frac(1, 4)$]. ] #math.equation(block: true, alt: "y equals −4 x squared plus 12 x minus 5")[$y = −4 x^(2) + 12 x − 5$] #math.equation(block: true, alt: "y equals x squared minus 6 x plus 15")[$y = x^(2) − 6 x + 15$] #solutionbox[ The minimum value is 6 when #math.equation(block: false, alt: "x equals 3")[$x = 3$]. ] #math.equation(block: true, alt: "y equals − x squared plus 4 x minus 5")[$y = "−" x^(2) + 4 x − 5$] #math.equation(block: true, alt: "y equals −9 x squared plus 16")[$y = −9 x^(2) + 16$] #solutionbox[ The maximum value is 16 when #math.equation(block: false, alt: "x equals 0")[$x = 0$]. ] #math.equation(block: true, alt: "y equals 4 x squared minus 49")[$y = 4 x^(2) − 49$] In the following exercises, solve. Round answers to the nearest tenth. An arrow is shot vertically upward from a platform 45 feet high at a rate of 168 ft/sec. Use the quadratic equation #math.equation(block: false, alt: "h equals −16 t squared plus 168 t plus 45")[$h = −16 t^(2) + 168 t + 45$] to find how long it will take the arrow to reach its maximum height, and then find the maximum height. #solutionbox[ In 5.3 sec the arrow will reach maximum height of 486 ft. ] A stone is thrown vertically upward from a platform that is 20 feet high at a rate of 160 ft/sec. Use the quadratic equation #math.equation(block: false, alt: "h equals −16 t squared plus 160 t plus 20")[$h = −16 t^(2) + 160 t + 20$] to find how long it will take the stone to reach its maximum height, and then find the maximum height. A computer store owner estimates that by charging #math.equation(block: false, alt: "x")[$x$] dollars each for a certain computer, he can sell #math.equation(block: false, alt: "40 minus x")[$40 − x$] computers each week. The quadratic equation #math.equation(block: false, alt: "R equals − x squared plus 40 x")[$R = "−" x^(2) + 40 x$] is used to find the revenue, #math.equation(block: false, alt: "R")[$R$], received when the selling price of a computer is #math.equation(block: false, alt: "x")[$x$]. Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue. #solutionbox[ Charging \$20 for each computer will give the maximum revenue of \$400. ] A retailer who sells backpacks estimates that, by selling them for #math.equation(block: false, alt: "x")[$x$] dollars each, he will be able to sell #math.equation(block: false, alt: "100 minus x")[$100 − x$] backpacks a month. The quadratic equation #math.equation(block: false, alt: "R equals − x squared plus 100 x")[$R = "−" x^(2) + 100 x$] is used to find the #math.equation(block: false, alt: "R")[$R$] received when the selling price of a backpack is #math.equation(block: false, alt: "x")[$x$]. Find the selling price that will give him the maximum revenue, and then find the amount of the maximum revenue. A rancher is going to fence three sides of a corral next to a river. He needs to maximize the corral area using 240 feet of fencing. The quadratic equation #math.equation(block: false, alt: "A equals x open parenthesis 240 minus 2 x close parenthesis")[$A = x ( 240 − 2 x )$] gives the area of the corral, #math.equation(block: false, alt: "A")[$A$], for the length, #math.equation(block: false, alt: "x ,")[$x ,$] of the corral along the river. Find the length of the corral along the river that will give the maximum area, and then find the maximum area of the corral. #solutionbox[ The length of the side along the river of the corral is 60 feet and the maximum area is 7,200 sq ft. ] A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation #math.equation(block: false, alt: "A equals x open parenthesis 100 minus 2 x close parenthesis")[$A = x ( 100 − 2 x )$] gives the area, #math.equation(block: false, alt: "A")[$A$], of the dog run for the length, #math.equation(block: false, alt: "x")[$x$], of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run. ==== Everyday Math In the previous set of exercises, you worked with the quadratic equation #math.equation(block: false, alt: "R equals − x squared plus 40 x")[$R = "−" x^(2) + 40 x$] that modeled the revenue received from selling computers at a price of #math.equation(block: false, alt: "x")[$x$] dollars. You found the selling price that would give the maximum revenue and calculated the maximum revenue. Now you will look at more characteristics of this model. #linebreak() ⓐ Graph the equation #math.equation(block: false, alt: "R equals − x squared plus 40 x")[$R = "−" x^(2) + 40 x$]. ⓑ Find the values of the #emph[x]-intercepts. #solutionbox[ + ⓐ #linebreak() #figure(figph[This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 60. The y-axis of the plane runs from -50 to 500. The parabola has a vertex at (20, 400) and also goes through the points (0, 0) and (40, 0).], alt: "This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 60. The y-axis of the plane runs from -50 to 500. The parabola has a vertex at (20, 400) and also goes through the points (0, 0) and (40, 0).", caption: none) + ⓑ #math.equation(block: false, alt: "open parenthesis 0 , 0 close parenthesis , open parenthesis 40 , 0 close parenthesis")[$( 0 , 0 ) , ( 40 , 0 )$] ] In the previous set of exercises, you worked with the quadratic equation #math.equation(block: false, alt: "R equals − x squared plus 100 x")[$R = "−" x^(2) + 100 x$] that modeled the revenue received from selling backpacks at a price of #math.equation(block: false, alt: "x")[$x$] dollars. You found the selling price that would give the maximum revenue and calculated the maximum revenue. Now you will look at more characteristics of this model. #linebreak() ⓐ Graph the equation #math.equation(block: false, alt: "R equals − x squared plus 100 x")[$R = "−" x^(2) + 100 x$]. ⓑ Find the values of the #emph[x]-intercepts. ==== Writing Exercises For the revenue model in and , explain what the #emph[x]-intercepts mean to the computer store owner. #solutionbox[ Answers will vary. ] For the revenue model in and , explain what the #emph[x]-intercepts mean to the backpack retailer. ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[This table has six rows and four columns. The first row is a header row and it labels each column. The first column is labeled "I can …", the second "Confidently", the third “With some help” and the last "No–I don’t get it". In the “I can…” column the second row reads “solve quadratic equations using the quadratic for recognize the graph of a quadratic equation in two variables.” The third row reads “find the axis of symmetry and vertex of a parabola.” The fourth row reads “find the intercepts of a parabola.” The fifth row reads “graph quadratic equations in two variables.” and the last row reads “solve maximum and minimum applications.” The remaining columns are blank.], alt: "This table has six rows and four columns. The first row is a header row and it labels each column. The first column is labeled \"I can …\", the second \"Confidently\", the third “With some help” and the last \"No–I don’t get it\". In the “I can…” column the second row reads “solve quadratic equations using the quadratic for recognize the graph of a quadratic equation in two variables.” The third row reads “find the axis of symmetry and vertex of a parabola.” The fourth row reads “find the intercepts of a parabola.” The fifth row reads “graph quadratic equations in two variables.” and the last row reads “solve maximum and minimum applications.” The remaining columns are blank.", caption: none) ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve? === Chapter 10 Review Exercises ==== 10.1 Solve Quadratic Equations Using the Square Root Property In the following exercises, solve using the Square Root Property. #math.equation(block: true, alt: "x squared equals 100")[$x^(2) = 100$] #solutionbox[ #math.equation(block: true, alt: "x equals plus or minus 10")[$x = ± #h(0.2em) 10$] ] #math.equation(block: true, alt: "y squared equals 144")[$y^(2) = 144$] #math.equation(block: true, alt: "m squared minus 40 equals 0")[$m^(2) − 40 = 0$] #solutionbox[ #math.equation(block: true, alt: "m equals plus or minus 2 the square root of 10")[$m = ± #h(0.2em) 2 sqrt(10)$] ] #math.equation(block: true, alt: "n squared minus 80 equals 0")[$n^(2) − 80 = 0$] #math.equation(block: true, alt: "4 a squared equals 100")[$4 a^(2) = 100$] #solutionbox[ #math.equation(block: true, alt: "a equals plus or minus 5")[$a = ± #h(0.2em) 5$] ] #math.equation(block: true, alt: "2 b squared equals 72")[$2 b^(2) = 72$] #math.equation(block: true, alt: "r squared plus 32 equals 0")[$r^(2) + 32 = 0$] #solutionbox[ no solution ] #math.equation(block: true, alt: "t squared plus 18 equals 0")[$t^(2) + 18 = 0$] #math.equation(block: true, alt: "the fraction 4 over 3 v squared plus 4 equals 28")[$frac(4, 3) v^(2) + 4 = 28$] #solutionbox[ #math.equation(block: true, alt: "v equals plus or minus 3 the square root of 2")[$v = ± #h(0.2em) 3 sqrt(2)$] ] #math.equation(block: true, alt: "the fraction 2 over 3 w squared minus 20 equals 30")[$frac(2, 3) w^(2) − 20 = 30$] #math.equation(block: true, alt: "5 c squared plus 3 equals 19")[$5 c^(2) + 3 = 19$] #solutionbox[ #math.equation(block: true, alt: "c equals plus or minus the fraction 4 the square root of 5 over 5")[$c = ± #h(0.2em) frac(4 sqrt(5), 5)$] ] #math.equation(block: true, alt: "3 d squared minus 6 equals 43")[$3 d^(2) − 6 = 43$] In the following exercises, solve using the Square Root Property. #math.equation(block: true, alt: "open parenthesis p minus 5 close parenthesis squared plus 3 equals 19")[$attach(( p − 5 ), t: 2) + 3 = 19$] #solutionbox[ #math.equation(block: true, alt: "p equals 1 , 9")[$p = 1 , 9$] ] #math.equation(block: true, alt: "open parenthesis q plus 4 close parenthesis squared equals 9")[$attach(( q + 4 ), t: 2) = 9$] #math.equation(block: true, alt: "open parenthesis u plus 1 close parenthesis squared equals 45")[$attach(( u + 1 ), t: 2) = 45$] #solutionbox[ #math.equation(block: true, alt: "u equals −1 plus or minus 3 the square root of 5")[$u = −1 ± 3 sqrt(5)$] ] #math.equation(block: true, alt: "open parenthesis z minus 5 close parenthesis squared equals 50")[$attach(( z − 5 ), t: 2) = 50$] #math.equation(block: true, alt: "open parenthesis x minus the fraction 1 over 4 close parenthesis squared equals the fraction 3 over 16")[$attach(( x − frac(1, 4) ), t: 2) = frac(3, 16)$] #solutionbox[ #math.equation(block: true, alt: "x equals the fraction 1 over 4 plus or minus the fraction the square root of 3 over 4")[$x = frac(1, 4) ± frac(sqrt(3), 4)$] ] #math.equation(block: true, alt: "open parenthesis y minus the fraction 2 over 3 close parenthesis squared equals the fraction 2 over 9")[$attach(( y − frac(2, 3) ), t: 2) = frac(2, 9)$] #math.equation(block: true, alt: "open parenthesis m minus 7 close parenthesis squared plus 6 equals 30")[$attach(( m − 7 ), t: 2) + 6 = 30$] #solutionbox[ #math.equation(block: true, alt: "m equals 7 plus or minus 2 the square root of 6")[$m = 7 ± 2 sqrt(6)$] ] #math.equation(block: true, alt: "open parenthesis n minus 4 close parenthesis squared minus 50 equals 150")[$attach(( n − 4 ), t: 2) − 50 = 150$] #math.equation(block: true, alt: "open parenthesis 5 c plus 3 close parenthesis squared equals −20")[$attach(( 5 c + 3 ), t: 2) = −20$] #solutionbox[ no solution ] #math.equation(block: true, alt: "open parenthesis 4 c minus 1 close parenthesis squared equals −18")[$attach(( 4 c − 1 ), t: 2) = −18$] #math.equation(block: true, alt: "m squared minus 6 m plus 9 equals 48")[$m^(2) − 6 m + 9 = 48$] #solutionbox[ #math.equation(block: true, alt: "m equals 3 plus or minus 4 the square root of 3")[$m = 3 ± 4 sqrt(3)$] ] #math.equation(block: true, alt: "n squared plus 10 n plus 25 equals 12")[$n^(2) + 10 n + 25 = 12$] #math.equation(block: true, alt: "64 a squared plus 48 a plus 9 equals 81")[$64 a^(2) + 48 a + 9 = 81$] #solutionbox[ #math.equation(block: true, alt: "a equals minus the fraction 3 over 2 , the fraction 3 over 4")[$a = − frac(3, 2) , frac(3, 4)$] ] #math.equation(block: true, alt: "4 b squared minus 28 b plus 49 equals 25")[$4 b^(2) − 28 b + 49 = 25$] ==== 10.2 Solve Quadratic Equations Using Completing the Square In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared. #math.equation(block: true, alt: "x squared plus 22 x")[$x^(2) + 22 x$] #solutionbox[ #math.equation(block: true, alt: "open parenthesis x plus 11 close parenthesis squared")[$attach(( x + 11 ), t: 2)$] ] #math.equation(block: true, alt: "y squared plus 6 y")[$y^(2) + 6 y$] #math.equation(block: true, alt: "m squared minus 8 m")[$m^(2) − 8 m$] #solutionbox[ #math.equation(block: true, alt: "open parenthesis m minus 4 close parenthesis squared")[$attach(( m − 4 ), t: 2)$] ] #math.equation(block: true, alt: "n squared minus 10 n")[$n^(2) − 10 n$] #math.equation(block: true, alt: "a squared minus 3 a")[$a^(2) − 3 a$] #solutionbox[ #math.equation(block: true, alt: "open parenthesis a minus the fraction 3 over 2 close parenthesis squared")[$attach(( a − frac(3, 2) ), t: 2)$] ] #math.equation(block: true, alt: "b squared plus 13 b")[$b^(2) + 13 b$] #math.equation(block: true, alt: "p squared plus the fraction 4 over 5 p")[$p^(2) + frac(4, 5) p$] #solutionbox[ #math.equation(block: true, alt: "open parenthesis p plus the fraction 2 over 5 close parenthesis squared")[$attach(( p + frac(2, 5) ), t: 2)$] ] #math.equation(block: true, alt: "q squared minus the fraction 1 over 3 q")[$q^(2) − frac(1, 3) q$] In the following exercises, solve by completing the square. #math.equation(block: true, alt: "c squared plus 20 c equals 21")[$c^(2) + 20 c = 21$] #solutionbox[ #math.equation(block: true, alt: "c equals 1 , −21")[$c = 1 , −21$] ] #math.equation(block: true, alt: "d squared plus 14 d equals −13")[$d^(2) + 14 d = −13$] #math.equation(block: true, alt: "x squared minus 4 x equals 32")[$x^(2) − 4 x = 32$] #solutionbox[ #math.equation(block: true, alt: "x equals −4 , 8")[$x = −4 , 8$] ] #math.equation(block: true, alt: "y squared minus 16 y equals 36")[$y^(2) − 16 y = 36$] #math.equation(block: true, alt: "r squared plus 6 r equals −100")[$r^(2) + 6 r = −100$] #solutionbox[ no solution ] #math.equation(block: true, alt: "t squared minus 12 t equals −40")[$t^(2) − 12 t = −40$] #math.equation(block: true, alt: "v squared minus 14 v equals −31")[$v^(2) − 14 v = −31$] #solutionbox[ #math.equation(block: true, alt: "v equals 7 plus or minus 3 the square root of 2")[$v = 7 ± 3 sqrt(2)$] ] #math.equation(block: true, alt: "w squared minus 20 w equals 100")[$w^(2) − 20 w = 100$] #math.equation(block: true, alt: "m squared plus 10 m minus 4 equals −13")[$m^(2) + 10 m − 4 = −13$] #solutionbox[ #math.equation(block: true, alt: "m equals −9 , −1")[$m = −9 , −1$] ] #math.equation(block: true, alt: "n squared minus 6 n plus 11 equals 34")[$n^(2) − 6 n + 11 = 34$] #math.equation(block: true, alt: "a squared equals 3 a plus 8")[$a^(2) = 3 a + 8$] #solutionbox[ #math.equation(block: true, alt: "a equals the fraction 3 over 2 plus or minus the fraction the square root of 41 over 2")[$a = frac(3, 2) ± frac(sqrt(41), 2)$] ] #math.equation(block: true, alt: "b squared equals 11 b minus 5")[$b^(2) = 11 b − 5$] #math.equation(block: true, alt: "open parenthesis u plus 8 close parenthesis open parenthesis u plus 4 close parenthesis equals 14")[$( u + 8 ) ( u + 4 ) = 14$] #solutionbox[ #math.equation(block: true, alt: "u equals −6 plus or minus 3 the square root of 2")[$u = −6 ± 3 sqrt(2)$] ] #math.equation(block: true, alt: "open parenthesis z minus 10 close parenthesis open parenthesis z plus 2 close parenthesis equals 28")[$( z − 10 ) ( z + 2 ) = 28$] #math.equation(block: true, alt: "3 p squared minus 18 p plus 15 equals 15")[$3 p^(2) − 18 p + 15 = 15$] #solutionbox[ #math.equation(block: true, alt: "p equals 0 , 6")[$p = 0 , 6$] ] #math.equation(block: true, alt: "5 q squared plus 70 q plus 20 equals 0")[$5 q^(2) + 70 q + 20 = 0$] #math.equation(block: true, alt: "4 y squared minus 6 y equals 4")[$4 y^(2) − 6 y = 4$] #solutionbox[ #math.equation(block: true, alt: "y equals minus the fraction 1 over 2 , 2")[$y = − frac(1, 2) , 2$] ] #math.equation(block: true, alt: "2 x squared plus 2 x equals 4")[$2 x^(2) + 2 x = 4$] #math.equation(block: true, alt: "3 c squared plus 2 c equals 9")[$3 c^(2) + 2 c = 9$] #solutionbox[ #math.equation(block: true, alt: "c equals minus the fraction 1 over 3 plus or minus the fraction 2 the square root of 7 over 3")[$c = − frac(1, 3) ± frac(2 sqrt(7), 3)$] ] #math.equation(block: true, alt: "4 d squared minus 2 d equals 8")[$4 d^(2) − 2 d = 8$] ==== 10.3 Solve Quadratic Equations Using the Quadratic Formula In the following exercises, solve by using the Quadratic Formula. #math.equation(block: true, alt: "4 x squared minus 5 x plus 1 equals 0")[$4 x^(2) − 5 x + 1 = 0$] #solutionbox[ #math.equation(block: true, alt: "x equals the fraction 1 over 4 , 1")[$x = frac(1, 4) , 1$] ] #math.equation(block: true, alt: "7 y squared plus 4 y minus 3 equals 0")[$7 y^(2) + 4 y − 3 = 0$] #math.equation(block: true, alt: "r squared minus r minus 42 equals 0")[$r^(2) − r − 42 = 0$] #solutionbox[ #math.equation(block: true, alt: "r equals −6 , 7")[$r = −6 , 7$] ] #math.equation(block: true, alt: "t squared plus 13 t plus 22 equals 0")[$t^(2) + 13 t + 22 = 0$] #math.equation(block: true, alt: "4 v squared plus v minus 5 equals 0")[$4 v^(2) + v − 5 = 0$] #solutionbox[ #math.equation(block: true, alt: "v equals minus the fraction 5 over 4 , 1")[$v = − frac(5, 4) , 1$] ] #math.equation(block: true, alt: "2 w squared plus 9 w plus 2 equals 0")[$2 w^(2) + 9 w + 2 = 0$] #math.equation(block: true, alt: "3 m squared plus 8 m plus 2 equals 0")[$3 m^(2) + 8 m + 2 = 0$] #solutionbox[ #math.equation(block: true, alt: "m equals the fraction −4 plus or minus the square root of 10 over 3")[$m = frac(−4 ± sqrt(10), 3)$] ] #math.equation(block: true, alt: "5 n squared plus 2 n minus 1 equals 0")[$5 n^(2) + 2 n − 1 = 0$] #math.equation(block: true, alt: "6 a squared minus 5 a plus 2 equals 0")[$6 a^(2) − 5 a + 2 = 0$] #solutionbox[ no real solution ] #math.equation(block: true, alt: "4 b squared minus b plus 8 equals 0")[$4 b^(2) − b + 8 = 0$] #math.equation(block: true, alt: "u open parenthesis u minus 10 close parenthesis plus 3 equals 0")[$u ( u − 10 ) + 3 = 0$] #solutionbox[ #math.equation(block: true, alt: "u equals 5 plus or minus the square root of 22")[$u = 5 ± sqrt(22)$] ] #math.equation(block: true, alt: "5 z open parenthesis z minus 2 close parenthesis equals 3")[$5 z ( z − 2 ) = 3$] #math.equation(block: true, alt: "the fraction 1 over 8 p squared minus the fraction 1 over 5 p equals minus the fraction 1 over 20")[$frac(1, 8) p^(2) − frac(1, 5) p = − frac(1, 20)$] #solutionbox[ #math.equation(block: true, alt: "p equals the fraction 4 plus or minus the square root of 6 over 5")[$p = frac(4 ± sqrt(6), 5)$] ] #math.equation(block: true, alt: "the fraction 2 over 5 q squared plus the fraction 3 over 10 q equals the fraction 1 over 10")[$frac(2, 5) q^(2) + frac(3, 10) q = frac(1, 10)$] #math.equation(block: true, alt: "4 c squared plus 4 c plus 1 equals 0")[$4 c^(2) + 4 c + 1 = 0$] #solutionbox[ #math.equation(block: true, alt: "c equals minus the fraction 1 over 2")[$c = − frac(1, 2)$] ] #math.equation(block: true, alt: "9 d squared minus 12 d equals −4")[$9 d^(2) − 12 d = −4$] In the following exercises, determine the number of solutions to each quadratic equation. + ⓐ #math.equation(block: false, alt: "9 x squared minus 6 x plus 1 equals 0")[$9 x^(2) − 6 x + 1 = 0$] + ⓑ #math.equation(block: false, alt: "3 y squared minus 8 y plus 1 equals 0")[$3 y^(2) − 8 y + 1 = 0$] + ⓒ #math.equation(block: false, alt: "7 m squared plus 12 m plus 4 equals 0")[$7 m^(2) + 12 m + 4 = 0$] + ⓓ #math.equation(block: false, alt: "5 n squared minus n plus 1 equals 0")[$5 n^(2) − n + 1 = 0$] #solutionbox[ ⓐ 1 ⓑ 2 ⓒ 2 ⓓ none ] + ⓐ #math.equation(block: false, alt: "5 x squared minus 7 x minus 8 equals 0")[$5 x^(2) − 7 x − 8 = 0$] + ⓑ #math.equation(block: false, alt: "7 x squared minus 10 x plus 5 equals 0")[$7 x^(2) − 10 x + 5 = 0$] + ⓒ #math.equation(block: false, alt: "25 x squared minus 90 x plus 81 equals 0")[$25 x^(2) − 90 x + 81 = 0$] + ⓓ #math.equation(block: false, alt: "15 x squared minus 8 x plus 4 equals 0")[$15 x^(2) − 8 x + 4 = 0$] In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. + ⓐ #math.equation(block: false, alt: "16 r squared minus 8 r plus 1 equals 0")[$16 r^(2) − 8 r + 1 = 0$] + ⓑ #math.equation(block: false, alt: "5 t squared minus 8 t plus 3 equals 9")[$5 t^(2) − 8 t + 3 = 9$] + ⓒ #math.equation(block: false, alt: "3 open parenthesis c plus 2 close parenthesis squared equals 15")[$3 attach(( c + 2 ), t: 2) = 15$] #solutionbox[ ⓐ factor ⓑ Quadratic Formula ⓒ square root ] + ⓐ #math.equation(block: false, alt: "4 d squared plus 10 d minus 5 equals 21")[$4 d^(2) + 10 d − 5 = 21$] + ⓑ #math.equation(block: false, alt: "25 x squared minus 60 x plus 36 equals 0")[$25 x^(2) − 60 x + 36 = 0$] + ⓒ #math.equation(block: false, alt: "6 open parenthesis 5 v minus 7 close parenthesis squared equals 150")[$6 attach(( 5 v − 7 ), t: 2) = 150$] ==== 10.4 Solve Applications Modeled by Quadratic Equations In the following exercises, solve by using methods of factoring, the square root principle, or the quadratic formula. Find two consecutive odd numbers whose product is 323. #solutionbox[ Two consecutive odd numbers whose product is 323 are 17 and 19, and #math.equation(block: false, alt: "−17")[$−17$] and #math.equation(block: false, alt: "−19 .")[$−19 .$] ] Find two consecutive even numbers whose product is 624. A triangular banner has an area of 351 square centimeters. The length of the base is two centimeters longer than four times the height. Find the height and length of the base. #solutionbox[ The height of the banner is 13 cm and the length of the side is 54 cm. ] Julius built a triangular display case for his coin collection. The height of the display case is six inches less than twice the width of the base. The area of the of the back of the case is 70 square inches. Find the height and width of the case. A tile mosaic in the shape of a right triangle is used as the corner of a rectangular pathway. The hypotenuse of the mosaic is 5 feet. One side of the mosaic is twice as long as the other side. What are the lengths of the sides? Round to the nearest tenth. #figure(figph[The image shows a rectangular pathway with a right inlaid in the lower left corner. The right angle of the triangle overlays the lower left corner of the rectangle. The left leg of the right triangle overlays the left side of the rectangle and the hypotenuse of the right triangle runs from the upper left corner of the rectangle to a point on the bottom of the rectangle.], alt: "The image shows a rectangular pathway with a right inlaid in the lower left corner. The right angle of the triangle overlays the lower left corner of the rectangle. The left leg of the right triangle overlays the left side of the rectangle and the hypotenuse of the right triangle runs from the upper left corner of the rectangle to a point on the bottom of the rectangle.", caption: none) #solutionbox[ The lengths of the sides of the mosaic are 2.2 and 4.4 feet. ] A rectangular piece of plywood has a diagonal which measures two feet more than the width. The length of the plywood is twice the width. What is the length of the plywood’s diagonal? Round to the nearest tenth. The front walk from the street to Pam’s house has an area of 250 square feet. Its length is two less than four times its width. Find the length and width of the sidewalk. Round to the nearest tenth. #solutionbox[ The width of the front walk is 8.2 feet and its length is 30.6 feet. ] For Sophia’s graduation party, several tables of the same width will be arranged end to end to give a serving table with a total area of 75 square feet. The total length of the tables will be two more than three times the width. Find the length and width of the serving table so Sophia can purchase the correct size tablecloth. Round answer to the nearest tenth. #figure(figph[The image shows four rectangular tables placed side by side to create one large table.], alt: "The image shows four rectangular tables placed side by side to create one large table.", caption: none) A ball is thrown vertically in the air with a velocity of 160 ft/sec. Use the formula #math.equation(block: false, alt: "h equals −16 t squared plus v sub 0 t")[$h = −16 t^(2) + v_(0) t$] to determine when the ball will be 384 feet from the ground. Round to the nearest tenth. #solutionbox[ The ball will reach 384 feet on its way up in 4 seconds and on the way down in 6 seconds. ] A bullet is fired straight up from the ground at a velocity of 320 ft/sec. Use the formula #math.equation(block: false, alt: "h equals −16 t squared plus v sub 0 t")[$h = −16 t^(2) + v_(0) t$] to determine when the bullet will reach 800 feet. Round to the nearest tenth. ==== 10.5 Graphing Quadratic Equations in Two Variables In the following exercises, graph by plotting point. Graph #math.equation(block: false, alt: "y equals x squared minus 2")[$y = x^(2) − 2$] #solutionbox[ #figure(figph[This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has a vertex at (0, -2) and goes through the point (1, -1).], alt: "This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has a vertex at (0, -2) and goes through the point (1, -1).", caption: none) ] Graph #math.equation(block: false, alt: "y equals − x squared plus 3")[$y = "−" x^(2) + 3$] In the following exercises, determine if the following parabolas open up or down. #math.equation(block: true, alt: "y equals −3 x squared plus 3 x minus 1")[$y = −3 x^(2) + 3 x − 1$] #solutionbox[ down ] #math.equation(block: true, alt: "y equals 5 x squared plus 6 x plus 3")[$y = 5 x^(2) + 6 x + 3$] #math.equation(block: true, alt: "y equals x squared plus 8 x minus 1")[$y = x^(2) + 8 x − 1$] #solutionbox[ up ] #math.equation(block: true, alt: "y equals −4 x squared minus 7 x plus 1")[$y = −4 x^(2) − 7 x + 1$] In the following exercises, find ⓐ the axis of symmetry and ⓑ the vertex. #math.equation(block: true, alt: "y equals − x squared plus 6 x plus 8")[$y = "−" x^(2) + 6 x + 8$] #solutionbox[ ⓐ #math.equation(block: false, alt: "x equals 3")[$x = 3$] ⓑ #math.equation(block: false, alt: "open parenthesis 3 , 17 close parenthesis")[$( 3 , 17 )$] ] #math.equation(block: true, alt: "y equals 2 x squared minus 8 x plus 1")[$y = 2 x^(2) − 8 x + 1$] In the following exercises, find the #emph[x]- and #emph[y]-intercepts. #math.equation(block: true, alt: "y equals x squared minus 4 x minus 5")[$y = x^(2) − 4 x − 5$] #solutionbox[ #math.equation(block: true, alt: "y : open parenthesis 0 , 5 close parenthesis ; x : open parenthesis 5 , 0 close parenthesis , open parenthesis −1 , 0 close parenthesis")[$y ":" #h(0.2em) ( 0 , 5 ) ; x ":" #h(0.2em) ( 5 , 0 ) , ( −1 , 0 )$] ] #math.equation(block: true, alt: "y equals x squared minus 8 x plus 15")[$y = x^(2) − 8 x + 15$] #math.equation(block: true, alt: "y equals x squared minus 4 x plus 10")[$y = x^(2) − 4 x + 10$] #solutionbox[ #math.equation(block: true, alt: "y : open parenthesis 0 , 10 close parenthesis ; x : none")[$y ":" #h(0.2em) ( 0 , 10 ) ; x ":" #h(0.2em) "none"$] ] #math.equation(block: true, alt: "y equals −5 x squared minus 30 x minus 46")[$y = −5 x^(2) − 30 x − 46$] #math.equation(block: true, alt: "y equals 16 x squared minus 8 x plus 1")[$y = 16 x^(2) − 8 x + 1$] #solutionbox[ #math.equation(block: true, alt: "y : open parenthesis 0 , 1 close parenthesis ; x : open parenthesis the fraction 1 over 4 , 0 close parenthesis")[$y ":" #h(0.2em) ( 0 , 1 ) ; x ":" #h(0.2em) ( frac(1, 4) , 0 )$] ] #math.equation(block: true, alt: "y equals x squared plus 16 x plus 64")[$y = x^(2) + 16 x + 64$] In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry. #math.equation(block: true, alt: "y equals x squared plus 8 x plus 15")[$y = x^(2) + 8 x + 15$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , 15 close parenthesis ; x : open parenthesis −3 , 0 close parenthesis , open parenthesis −5 , 0 close parenthesis ;")[$y ":" #h(0.2em) ( 0 , 15 ) ; x ":" #h(0.2em) ( −3 , 0 ) , ( −5 , 0 ) ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals −4 ; vertex : open parenthesis −4 , −1 close parenthesis")[$x = −4 ; "vertex" ":" #h(0.2em) ( −4 , −1 )$] #linebreak() #figure(figph[This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -2 to 17. The parabola has points plotted at the vertex (-4, -1) and the intercepts (-3, 0), (-5, 0) and (0, 15). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -4.], alt: "This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -2 to 17. The parabola has points plotted at the vertex (-4, -1) and the intercepts (-3, 0), (-5, 0) and (0, 15). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -4.", caption: none) ] #math.equation(block: true, alt: "y equals x squared minus 2 x minus 3")[$y = x^(2) − 2 x − 3$] #math.equation(block: true, alt: "y equals − x squared plus 8 x minus 16")[$y = "−" x^(2) + 8 x − 16$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , −16 close parenthesis ; x : open parenthesis 4 , 0 close parenthesis ;")[$y ":" #h(0.2em) ( 0 , −16 ) ; x ":" #h(0.2em) ( 4 , 0 ) ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals 4 ; vertex : open parenthesis 4 , 0 close parenthesis")[$x = 4 ; "vertex" ":" #h(0.2em) ( 4 , 0 )$] #linebreak() #figure(figph[This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -15 to 12. The y-axis of the plane runs from -20 to 2. The parabola has points plotted at the vertex (4, 0) and the intercept (0, -16). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 4.], alt: "This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -15 to 12. The y-axis of the plane runs from -20 to 2. The parabola has points plotted at the vertex (4, 0) and the intercept (0, -16). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 4.", caption: none) ] #math.equation(block: true, alt: "y equals 4 x squared minus 4 x plus 1")[$y = 4 x^(2) − 4 x + 1$] #math.equation(block: true, alt: "y equals x squared plus 6 x plus 13")[$y = x^(2) + 6 x + 13$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , 13 close parenthesis ; x : none ;")[$y ":" #h(0.2em) ( 0 , 13 ) ; x ":" #h(0.2em) "none" ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals −3 ; vertex : open parenthesis −3 , 4 close parenthesis")[$x = −3 ; "vertex" ":" #h(0.2em) ( −3 , 4 )$] #linebreak() #figure(figph[This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -2 to 18. The parabola has points plotted at the vertex (-3, 4) and the intercept (0, 13). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -3.], alt: "This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -2 to 18. The parabola has points plotted at the vertex (-3, 4) and the intercept (0, 13). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -3.", caption: none) ] #math.equation(block: true, alt: "y equals −2 x squared minus 8 x minus 12")[$y = −2 x^(2) − 8 x − 12$] #math.equation(block: true, alt: "y equals −4 x squared plus 16 x minus 11")[$y = −4 x^(2) + 16 x − 11$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , −11 close parenthesis x : open parenthesis 3.1 , 0 close parenthesis , open parenthesis 0.9 , 0 close parenthesis ;")[$y ":" #h(0.2em) ( 0 , −11 ) #h(0.2em) x ":" #h(0.2em) ( 3.1 , 0 ) , ( 0.9 , 0 ) ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals 2 ; vertex: open parenthesis 2 , 5 close parenthesis")[$x = 2 ; #h(0.2em) "vertex:" #h(0.2em) ( 2 , 5 )$] #linebreak() #figure(figph[This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (2, 5) and the intercepts (3.1, 0) and (0.9, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 2.], alt: "This figure shows a downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (2, 5) and the intercepts (3.1, 0) and (0.9, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 2.", caption: none) ] #math.equation(block: true, alt: "y equals x squared plus 8 x plus 10")[$y = x^(2) + 8 x + 10$] In the following exercises, find the minimum or maximum value. #math.equation(block: true, alt: "y equals 7 x squared plus 14 x plus 6")[$y = 7 x^(2) + 14 x + 6$] #solutionbox[ The minimum value is #math.equation(block: false, alt: "−1")[$−1$] when #math.equation(block: false, alt: "x equals −1")[$x = −1$]. ] #math.equation(block: true, alt: "y equals −3 x squared plus 12 x minus 10")[$y = −3 x^(2) + 12 x − 10$] In the following exercises, solve. Rounding answers to the nearest tenth. A ball is thrown upward from the ground with an initial velocity of 112 ft/sec. Use the quadratic equation #math.equation(block: false, alt: "h equals −16 t squared plus 112 t")[$h = −16 t^(2) + 112 t$] to find how long it will take the ball to reach maximum height, and then find the maximum height. #solutionbox[ In 3.5 seconds the ball is at its maximum height of 196 feet. ] A daycare facility is enclosing a rectangular area along the side of their building for the children to play outdoors. They need to maximize the area using 180 feet of fencing on three sides of the yard. The quadratic equation #math.equation(block: false, alt: "A equals −2 x squared plus 180 x")[$A = −2 x^(2) + 180 x$] gives the area, #math.equation(block: false, alt: "A")[$A$], of the yard for the length, #math.equation(block: false, alt: "x")[$x$], of the building that will border the yard. Find the length of the building that should border the yard to maximize the area, and then find the maximum area. === Practice Test Use the Square Root Property to solve the quadratic equation: #math.equation(block: false, alt: "3 open parenthesis w plus 5 close parenthesis squared equals 27")[$3 attach(( w + 5 ), t: 2) = 27$]. #solutionbox[ #math.equation(block: true, alt: "w equals −2 , −8")[$w = −2 , −8$] ] Use Completing the Square to solve the quadratic equation: #math.equation(block: false, alt: "a squared minus 8 a plus 7 equals 23")[$a^(2) − 8 a + 7 = 23$]. Use the Quadratic Formula to solve the quadratic equation: #math.equation(block: false, alt: "2 m squared minus 5 m plus 3 equals 0")[$2 m^(2) − 5 m + 3 = 0$]. #solutionbox[ #math.equation(block: true, alt: "m equals 1 , the fraction 3 over 2")[$m = 1 , frac(3, 2)$] ] Solve the following quadratic equations. Use any method. #math.equation(block: true, alt: "8 v squared plus 3 equals 35")[$8 v^(2) + 3 = 35$] #math.equation(block: true, alt: "3 n squared plus 8 n plus 3 equals 0")[$3 n^(2) + 8 n + 3 = 0$] #solutionbox[ #math.equation(block: true, alt: "n equals the fraction −4 plus or minus the square root of 7 over 3")[$n = frac(−4 ± sqrt(7), 3)$] ] #math.equation(block: true, alt: "2 b squared plus 6 b minus 8 equals 0")[$2 b^(2) + 6 b − 8 = 0$] #math.equation(block: true, alt: "x open parenthesis x plus 3 close parenthesis plus 12 equals 0")[$x ( x + 3 ) + 12 = 0$] #solutionbox[ no real solution ] #math.equation(block: true, alt: "the fraction 4 over 3 y squared minus 4 y plus 3 equals 0")[$frac(4, 3) y^(2) − 4 y + 3 = 0$] Use the discriminant to determine the number of solutions of each quadratic equation. #math.equation(block: true, alt: "6 p squared minus 13 p plus 7 equals 0")[$6 p^(2) − 13 p + 7 = 0$] #solutionbox[ 2 ] #math.equation(block: true, alt: "3 q squared minus 10 q plus 12 equals 0")[$3 q^(2) − 10 q + 12 = 0$] Solve by factoring, the Square Root Property, or the Quadratic Formula. Find two consecutive even numbers whose product is 360. #solutionbox[ Two consecutive even number are #math.equation(block: false, alt: "−20")[$−20$] and #math.equation(block: false, alt: "−18")[$−18$] and 18 and 20. ] The length of a diagonal of a rectangle is three more than the width. The length of the rectangle is three times the width. Find the length of the diagonal. (Round to the nearest tenth.) For each parabola, find ⓐ which ways it opens, ⓑ the axis of symmetry, ⓒ the vertex, ⓓ the #emph[x]- and #emph[y]-intercepts, and ⓔ the maximum or minimum value. #math.equation(block: true, alt: "y equals 3 x squared plus 6 x plus 8")[$y = 3 x^(2) + 6 x + 8$] #solutionbox[ ⓐ up ⓑ #math.equation(block: false, alt: "x equals −1")[$x = −1$] ⓒ #math.equation(block: false, alt: "open parenthesis −1 , 5 close parenthesis")[$( −1 , 5 )$] ⓓ #math.equation(block: false, alt: "y : open parenthesis 0 , 8 close parenthesis ; x : none")[$y ":" #h(0.2em) ( 0 , 8 ) ; x ":" #h(0.2em) "none"$] ⓔ minimum value of 5 when #math.equation(block: false, alt: "x equals −1")[$x = −1$] ] #math.equation(block: true, alt: "y equals x squared minus 4")[$y = x^(2) − 4$] #math.equation(block: true, alt: "y equals x squared plus 10 x plus 24")[$y = x^(2) + 10 x + 24$] #solutionbox[ ⓐ up ⓑ #math.equation(block: false, alt: "x equals −5")[$x = −5$] ⓒ #math.equation(block: false, alt: "open parenthesis −5 , −1 close parenthesis")[$( −5 , −1 )$] ⓓ #math.equation(block: false, alt: "y ; open parenthesis 0 , 24 close parenthesis ; x : open parenthesis −6 , 0 close parenthesis , open parenthesis −4 , 0 close parenthesis")[$y ; ( 0 , 24 ) ; x ":" #h(0.2em) ( −6 , 0 ) , ( −4 , 0 )$] ⓔ minimum value of #math.equation(block: false, alt: "−1")[$−1$] when #math.equation(block: false, alt: "x equals −5")[$x = −5$] ] #math.equation(block: true, alt: "y equals −3 x squared plus 12 x minus 8")[$y = −3 x^(2) + 12 x − 8$] #math.equation(block: true, alt: "y equals − x squared minus 8 x plus 16")[$y = "−" x^(2) − 8 x + 16$] #solutionbox[ ⓐ down ⓑ #math.equation(block: false, alt: "x equals −4")[$x = −4$] #linebreak() ⓒ #math.equation(block: false, alt: "open parenthesis −4 , 32 close parenthesis")[$( −4 , 32 )$] ⓓ #math.equation(block: false, alt: "y ; open parenthesis 0 , 16 close parenthesis ; x : open parenthesis −9.7 , 0 close parenthesis , open parenthesis 1.7 , 0 close parenthesis")[$y ; ( 0 , 16 ) ; x ":" #h(0.2em) ( −9.7 , 0 ) , ( 1.7 , 0 )$] #linebreak() ⓔ maximum value of #math.equation(block: false, alt: "32")[$32$] when #math.equation(block: false, alt: "x equals −4")[$x = −4$] ] Graph the following parabolas by using intercepts, the vertex, and the axis of symmetry. #math.equation(block: true, alt: "y equals 2 x squared plus 6 x plus 2")[$y = 2 x^(2) + 6 x + 2$] #math.equation(block: true, alt: "y equals 16 x squared plus 24 x plus 9")[$y = 16 x^(2) + 24 x + 9$] #solutionbox[ #math.equation(block: false, alt: "y : open parenthesis 0 , 9 close parenthesis ; x : open parenthesis minus the fraction 3 over 4 , 0 close parenthesis ;")[$y ":" #h(0.2em) ( 0 , 9 ) ; x ":" #h(0.2em) ( − frac(3, 4) , 0 ) ;$] #linebreak() axis: #math.equation(block: false, alt: "x equals minus the fraction 3 over 4 ; vertex : open parenthesis minus the fraction 3 over 4 , 0 close parenthesis")[$x = − frac(3, 4) ; "vertex" ":" #h(0.2em) ( − frac(3, 4) , 0 )$] #linebreak() #figure(figph[This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (3 fourths, 0) and the intercept (0, 9). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 3 fourths.], alt: "This figure shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The parabola has points plotted at the vertex (3 fourths, 0) and the intercept (0, 9). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 3 fourths.", caption: none) ] Solve. A water balloon is launched upward at the rate of 86 ft/sec. Using the formula #math.equation(block: false, alt: "h equals −16 t squared plus 86 t")[$h = −16 t^(2) + 86 t$], find how long it will take the balloon to reach the maximum height and then find the maximum height. Round to the nearest tenth.